Horizon Entropy Research Articles

Warped CFT duals of the Plebański-Demiański family of solutions

In this paper, we analyze the symmetry properties of the complete family of type D spacetimes generalized form the Plebański-Demiański solution in four dimensions holographically in terms of a warped CFT. The generalized Plebański-Demiański solutions are black hole-like spacetimes characterized by seven physical parameters. Most of the black holes in four dimensions are included within this family. Generically consider a solution with horizon in this family, we figure out the possible warped conformal symmetry attached to the horizon. The horizon can be either extremal or non-extremal. In the extremal case, the near horizon region can be mapped to an infinite spacetime with geometry given by a warped and twist product of AdS2 and S2. The new boundary conditions for AdS2 as well as their higher dimensional uplifts are applied here to manifest the asymptotic symmetry as the warped conformal symmetry. In the non-extremal case, the global warped conformal symmetry is singled out by analyzing the scalar wave equation with constant frequency. The local warped conformal symmetries are represented by the charge algebra associated to the vector fields which preserve the scalar wave equation as well as its frequency. In defining the variation of the covariant charges, a proper counterterm is introduced for consistency conditions which is supposed to be suitable for all the solutions within the family. As a consistency check, the horizon entropy is reproduced by the entropy formula of the warped CFT by using its modular covariance and the central terms derived in the bulk spacetimes.

An observer’s measure of de Sitter entropy

The two-point correlation function of a massive field ⟨χ(τ)χ(0)⟩, measured along an observer’s worldline in de Sitter (dS), decays exponentially as τ → ∞. Meanwhile, every dS observer is surrounded by a horizon and the holographic interpretation of the horizon entropy SdS suggests that the correlation function should stop decaying, and start behaving erratically at late times. We find evidence for this expectation in Jackiw-Teitelboim gravity by finding a topologically nontrivial saddle, which is suppressed by e−SdS\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {e}^{-{S}_{\ extrm{dS}}} $$\\end{document}, and which gives a constant contribution to | ⟨χ(τ)χ(0)⟩ |2. This constant might have the interpretation of the late-time average of | ⟨χ(τ)χ(0)⟩ |2 over all microscopic theories that have the same low-energy effective description.

Remarks on 2D quantum cosmology

We consider two-dimensional quantum gravity endowed with a positive cosmological constant and coupled to a conformal field theory of large and positive central charge. We study cosmological properties at the classical and quantum level. We provide a complete ADM analysis of the classical phase space, revealing a family of either bouncing or big bang/crunch type cosmologies. At the quantum level, we solve the Wheeler-DeWitt equation exactly. In the semiclassical limit, we link the Wheeler-DeWitt state space to the classical phase space. Wavefunctionals of the Hartle-Hawking and Vilenkin type are identified, and we uncover a quantum version of the bouncing spacetime. We retrieve the Hartle-Hawking wavefunction from the disk path integral of timelike Liouville theory. To do so, we must select a particular contour in the space of complexified fields. The quantum information content of the big bang cosmology is discussed, and contrasted with the de Sitter horizon entropy as computed by a gravitational path integral over the two-sphere.

Look Beyond Additivity and Extensivity of Entropy for Black Hole and Cosmological Horizons.

We present a comparative analysis of the plethora of nonextensive and/or nonadditive entropies which go beyond the standard Boltzmann-Gibbs formulation. After defining the basic notions of additivity, extensivity, and composability, we discuss the properties of these entropies and their mutual relations, if they exist. The results are presented in two informative tables that are of strong interest to the gravity and cosmology community in the context of the recently intensively explored horizon entropies for black hole and cosmological models. Gravitational systems admit long-range interactions, which usually lead to a break of the standard additivity rule for thermodynamic systems composed of subsystems in Boltzmann-Gibbs thermodynamics. The features of additivity, extensivity, and composability are listed systematically. A brief discussion on the validity of the notion of equilibrium temperature for nonextensive systems is also presented.

Different Aspects of Entropic Cosmology

We provide a short review of the recent developments in entropic cosmology based on two thermodynamic laws of the apparent horizon, namely the first and the second laws of thermodynamics. The first law essentially provides the change in entropy of the apparent horizon during the cosmic evolution of the universe; in particular, it is expressed by TdS=−d(ρV)+WdV (where W is the work density and other quantities have their usual meanings). In this way, the first law actually links various theories of gravity with the entropy of the apparent horizon. This leads to a natural question—“What is the form of the horizon entropy corresponding to a general modified theory of gravity?”. The second law of horizon thermodynamics states that the change in total entropy (the sum of horizon entropy + matter fields’ entropy) with respect to cosmic time must be positive, where the matter fields behave like an open system characterised by a non-zero chemical potential. The second law of horizon thermodynamics importantly provides model-independent constraints on entropic parameters. Finally, we discuss the standpoint of entropic cosmology on inflation (or bounce), reheating and primordial gravitational waves from the perspective of a generalised entropy function.

Interior volume of the charged BTZ black holes

In Schwarzschild spacetime, Reinhart (1973) has shown the hypersurface rss=3M/2 (the subscript stands for “steady-state”) to be the maximal hypersurface. This steady-state radius rss plays a crucial role in defining and evaluating the interior volume of a black hole. In this article, we investigate various methods to compute the maximal interior volume of a charged BTZ black hole. We find that the presence of charge Q in a black hole introduces a “log” term in the metric as a result of which, an analytical solution for the volume does not exist. So we first compute the volume of the black hole for the limiting case when the charge Q is very small (i.e., Q≪1: Q is a dimensionless parameter in (2+1) dimensions) and then carry out a numerical analysis to solve for the volume for more generic values of the charge. We find that the volume grows monotonically with the advance time v. We further investigate the functional behavior of the entropy of a massless scalar field living on the maximal hypersurface of a near-extremal black hole. We show that this volume entropy exhibits a very different functional form from the horizon entropy.

Action, entropy and pair creation rate of charged black holes in de Sitter space

We compute and clarify the interpretation of the on-shell Euclidean action for Reissner-Nordström black holes in de Sitter space. We show the on-shell action is minus the sum of the black hole and cosmological horizon entropy for arbitrary mass and charge in any number of dimensions. This unifying expression helps to clear up a confusion about the Euclidean actions of extremal and ultracold black holes in de Sitter, as they can be understood as special cases of the general expression. We then use this result to estimate the probability for the pair creation of black holes with arbitrary mass and charge in an empty de Sitter background, by employing the formalism of constrained instantons. Finally, we suggest that the decay of charged de Sitter black holes is governed by the gradient flow of the entropy function and that, as a consequence, the regime of light, superradiant, rapid charge emission should describe the potential decay of extreme charged Nariai black holes to singular geometries.

The Fermionic Entanglement Entropy of the Vacuum State of a Schwarzschild Black Hole Horizon

AbstractWe define and analyze the fermionic entanglement entropy of a Schwarzschild black hole horizon for the regularized vacuum state of an observer at infinity. Using separation of variables and an integral representation of the Dirac propagator, the entanglement entropy is computed to be a prefactor times the number of occupied angular momentum modes on the event horizon.

Thermodynamics and Decay of de Sitter Vacuum

We discuss the consequences of the unique symmetry of de Sitter spacetime. This symmetry leads to the specific thermodynamic properties of the de Sitter vacuum, which produces a thermal bath for matter. de Sitter spacetime is invariant under the modified translations, r→r−eHta, where H is the Hubble parameter. For H→0, this symmetry corresponds to the conventional invariance of Minkowski spacetime under translations r→r−a. Due to this symmetry, all the comoving observers at any point of the de Sitter space perceive the de Sitter environment as the thermal bath with temperature T=H/π, which is twice as large as the Gibbons–Hawking temperature of the cosmological horizon. This temperature does not violate de Sitter symmetry and, thus, does not require the preferred reference frame, as distinct from the thermal state of matter, which violates de Sitter symmetry. This leads to the heat exchange between gravity and matter and to the instability of the de Sitter state towards the creation of matter, its further heating, and finally the decay of the de Sitter state. The temperature T=H/π determines different processes in the de Sitter environment that are not possible in the Minkowski vacuum, such as the process of ionization of an atom in the de Sitter environment. This temperature also determines the local entropy of the de Sitter vacuum state, and this allows us to calculate the total entropy of the volume inside the cosmological horizon. The result reproduces the Gibbons–Hawking area law, which is attributed to the cosmological horizon, Shor=4πKA, where K=1/(16πG). This supports the holographic properties of the cosmological event horizon. We extend the consideration of the local thermodynamics of the de Sitter state using the f(R) gravity. In this thermodynamics, the Ricci scalar curvature R and the effective gravitational coupling K are thermodynamically conjugate variables. The holographic connection between the bulk entropy of the Hubble volume and the surface entropy of the cosmological horizon remains the same but with the gravitational coupling K=df/dR. Such a connection takes place only in the 3+1 spacetime, where there is a special symmetry due to which the variables K and R have the same dimensionality. We also consider the lessons from de Sitter symmetry for the thermodynamics of black and white holes.

Quantum entanglement on black hole horizons in string theory and holography

We compute the exact one-loop partition function of ℤN orbifolds of Euclidean BTZ black hole with the aim to compute the entanglement entropy of the black hole horizon in string theory as a function of the mass and spin of the black hole and the AdS3 radius. We analyze the tachyonic contribution to the modular integrand for the partition function known for odd integers N > 1 and show that it admits an analytic continuation resulting in a finite answer for the modular integral in the physical region 0 < N ≤ 1. We discuss the flat space limit and the relevance of this computation for quantum gravity near black hole horizons and holography in relation to the thermal entropy.

Using the Kaniadakis horizon entropy in the presence of neutrinos to alleviate the Hubble and S8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ S_{8} $$\\end{document} tensions

The H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{0}$$\\end{document} tension stands as a prominent challenge in cosmology, serving as a primary driver for exploring alternative models of dark energy. Another tension arises from measurements of the S8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ S_{8} $$\\end{document} parameter, which is characterize the amplitude of matter fluctuations in the universe. In this study, we address the alleviation of both the Hubble tension and S8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ S_{8} $$\\end{document} tension by incorporating Kaniadakis horizon entropy. We investigate two scenarios to explore the impact of this entropy on cosmological parameters. In the first scenario, utilizing modified Friedmann equations through Kaniadakis entropy, we estimate the values of H0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H_{0}$$\\end{document} and S8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ S_{8} $$\\end{document}. In the subsequent scenario, we introduce the neutrino term and assess its effect on mitigating the Hubble and S8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ S_{8} $$\\end{document} tensions. Our findings reveal that when considering the first scenario, the results closely align with Planck’s 2018 outcomes for Hubble and S8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ S_{8} $$\\end{document} tensions. Moreover, with the inclusion of neutrinos, these tensions are alleviated to approximately 2σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma $$\\end{document}, and the S8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ S_{8} $$\\end{document} value is in full agreement with the results from the KiDS and DES survey. Furthermore, we impose a constraint on the parameter K in each scenario. Our analysis yields K=0.12±0.41\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K = 0.12\\pm 0.41$$\\end{document} for Kaniadakis entropy without neutrinos and K=0.39±0.4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K = 0.39\\pm 0.4$$\\end{document} for the combined dataset considering Kaniadakis entropy in the presence of neutrinos. We demonstrate that the value of K may be affected by neutrino mass, which can cause energy transfer between different parts of the universe and alter the Hubble parameter value.

Emergence of cosmic space and the maximization of horizon entropy

The spatial expansion of the universe can be described as the emergence of space with the progress of cosmic time, through a simple equation, ΔV=ΔtNsurf−Nbulk. This law of emergence suggested by Padmanabhan in the context of general relativity for a flat universe has been generalized by Sheykhi to Gauss Bonnet and Lovelock gravity for a universe with any spatial curvature. We investigate whether this generalized holographic equipartition effectively implies the maximization of horizon entropy. First, we obtain the constraints imposed by the maximization of horizon entropy in Einstein, Gauss Bonnet and Lovelock gravities for a universe with any spatial curvature. We then analyze the consistency of the law of emergence in Ahmad Sheykhi (2013), with these constraints obtained. Interestingly, both the law of emergence and the horizon entropy maximization demands an asymptotically de Sitter universe with ω≥−1. More specifically, when the degrees of freedom in the bulk (Nbulk) becomes equal to the degrees of freedom on the boundary surface (Nsurf), the universe attains a state of maximum horizon entropy. Thus, the law of emergence can be viewed as a tendency for maximizing the horizon entropy, even in a non flat universe. Our results points at the deep connection between the law of emergence and horizon thermodynamics, beyond Einstein gravity irrespective of the spatial curvature.

Everything you always wanted to know about how causal set theory can help with open questions in cosmology, but were afraid to ask

In this paper, we briefly survey open questions in cosmology that either have been or could potentially be addressed with the help of the causal set theory approach to quantum gravity. Our discussion includes topics ranging from dark matter and dark energy to primordial quantum fluctuations and horizon entropy. By putting together all in one place, these cosmologically relevant directions of work within causal set theory, we hope to impart a bird’s-eye view of this important research theme.

Entropic Inflation in Presence of Scalar Field

In spirit of the recently proposed four-parameter generalized entropy of apparent horizon, we investigate inflationary cosmology where the matter field inside of the horizon is dominated by a scalar field with a power law potential (i.e., the form of ϕn where ϕ is the scalar field under consideration). Actually without any matter inside of the horizon, the entropic cosmology leads to a de-Sitter spacetime, or equivalently, an eternal inflation with no exit. Thus in order to achieve a viable inflation, we consider a minimally coupled scalar field inside the horizon, and moreover, with the simplest quadratic potential. It is well known that the ϕ2 potential in standard scalar field cosmology is ruled out from inflationary perspective as it is not consistent with the recent Planck 2018 data; (here it may be mentioned that in the realm of “apparent horizon thermodynamics”, the standard scalar field cosmology is analogous to the case where the entropy of the apparent horizon is given by the Bekenstein–Hawking entropy). However, the story becomes different if the horizon entropy is of generalized entropic form, in which case, the effective energy density coming from the horizon entropy plays a significant role during the evolution of the universe. In particular, it turns out that in the context of generalized entropic cosmology, the ϕ2 potential indeed leads to a viable inflation (according to the Planck data) with a graceful exit, and thus the potential can be made back in the scene.

Emergence of cosmic space in Tsallis modified gravity from equilibrium and non-equilibrium thermodynamic perspective

In the context of Tsallis entropy, we explore the connection between the law of emergence and the thermodynamic laws from a more accurate non-equilibrium perspective. Here, the equilibrium Clausius relation does not conform to the standard energy-momentum conservation. Therefore, an effective gravitational coupling is introduced to rewrite the field equation similar to general relativity, and the corresponding generalized continuity equation is obtained. As a result, thermodynamic laws were modified with the non-equilibrium energy dissipation and entropy production terms, using which we derive the law of emergence. The investigation of the law of emergence and the entropy maximization principle with Tsallis entropy in the non-equilibrium perspective shows that both result in the same constraints as obtained in other gravity theories and the equilibrium context of Tsallis entropy, except for an additional constraint on the Tsallis parameter as a result of extra entropy production. Consequently, the thermodynamic interpretation of the expansion of the universe stays valid even with quantum corrections to the horizon entropy since the correction terms in Tsallis entropy can be treated as the quantum corrections to Bekenstein-Hawking entropy.

DS2 supergravity

We construct two-dimensional supergravity theories endowed with a positive cosmological constant, that admit de Sitter vacua. We consider the cases of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 as well as N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetry, and couple the supergravity to a superconformal field theory with the same amount of supersymmetry. Upon fixing a supersymmetric extension of the Weyl gauge, the theories are captured, at the quantum level, by supersymmetric extensions of timelike Liouville theory with N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 and N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetry respectively. The theories exhibit good ultraviolet properties and are amenable to a variety of techniques such as systematic loop expansions and, in the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 case, supersymmetric localization. Our constructions offer a novel path toward a precise treatment of the Euclidean gravitational path integral for de Sitter, and in turn, the Gibbons-Hawking entropy of the de Sitter horizon. We argue that the supersymmetric localization method applied to the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theory must receive contributions from boundary terms in configuration space. We also discuss how these theories overcome several obstructions that appear upon combining de Sitter space with supersymmetry.

Physical charges versus conformal invariance in unimodular gravity

There is a formulation of unimodular gravity (UG) in which the unimodular metric is obtained out of an unrestricted one as [Formula: see text]. This unrestricted metric is only defined up to a Weyl rescaling. Physical charges like ADM energy or horizon entropy, however, are not Weyl invariant in general. In this paper a Weyl invariant definition of some physical charges is given.

Entropy current and fluid-gravity duality in Gauss-Bonnet theory

Working within the approximation of small amplitude expansion, recently an entropy current has been constructed on the horizons of dynamical black hole solution in any higher derivative theory of gravity. In this note, we have dualized this horizon entropy current to a boundary entropy current in an asymptotically AdS black hole metric with a dual description in terms of dynamical fluids living on the AdS boundary. This boundary entropy current is constructed using a set of mapping functions relating each point on the horizon to a point on the boundary. We have applied our construction to black holes in Einstein-Gauss-Bonnet theory. We have seen that up to the first order in derivative expansion, Gauss-Bonnet terms do not add any extra corrections to fluid entropy as expected. However, at the second order in derivative expansion, the boundary current will non-trivially depend on how we choose our horizon to boundary map, which need not be expressible entirely in terms of fluid variables. So generically, the boundary entropy current generated by dualizing the horizon current will not admit a fluid dynamical description.

Thermal evolution and stability analysis of phenomenologically emergent dark energy model

The phenomenologically emergent dark energy (PEDE) model is a varying dark energy model with no extra degrees of freedom proposed by Li and Shafieloo (Astrophys J 883(1):L3, 2019) to alleviate the Hubble tension. The statistical consistency of the model has been discussed by many authors. Since the model depicts a phantom dark energy that increases with redshift, its cosmic evolution, particularly during the late phase, must be examined. We discover that the model’s Hubble and deceleration parameters display unusual behaviour in the future, which differs from varLambda CDM cosmology. We find the model also follows a distinct evolution in the statefinder plane. The phantom nature of the model leads to the violation of the null energy condition and a decrease in horizon entropy. The asymptotic future epoch also seems to be unstable based on our dynamical system analysis as well as the stability analysis based on dark energy sound speed.

Analog of the Sommerfeld Law in Quantum Vacuum

The activation temperature T in the de Sitter environment is twice the Gibbons–Hawking temperature, related to the cosmological horizon. We consider the activation temperature as the local temperature of the de Sitter vacuum, and construct the local thermodynamics of the de Sitter state. This thermodynamics includes also the gravitational coupling K and the scalar Riemann curvature mathcal{R} as the thermodynamically conjugate variables. These variables modify the thermodynamics of the Gibbs–Duhem relation in the de Sitter state. The free energy density is proportional to - {{T}^{2}}, which is similar to that in the nonrelativistic Fermi liquids and in relativistic matter with equation of state w = 1. The local entropy is proportional to the local temperature, while the total entropy inside the cosmological horizon is A{text{/}}4G, where A is the area of the horizon. This entropy is usually interpreted as the entropy of the cosmological horizon. We also consider the possible application of the de Sitter thermodynamics to the Schwarzschild–de Sitter black hole and to black and white holes with the de Sitter cores.