Thermodynamic extended phase space and P-V criticality of black holes at Pure Lovelock gravity

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In this work the chemistry of asymptotically AdS black hole, for the charged and uncharged solutions of Pure Lovelock gravity, is discussed. The charged case behaves as a Van der Waals fluid and whose first order phase transitions, between small stable/large stable black holes, are analogous to the liquid/gas phase transitions as in AdS black hole for Einstein Hilbert theory. However, the thermodynamics behavior differs from the generic Lovelock theory, because there is a unique critical point, unlike the generic case where there may be more than one critiackcal point. Also, it is shown that the thermodynamics behavior of the Pure Lovelock black holes (in the extended phase space) can be represented by variables that are analytic functions of n and d, where n corresponds to the highest power of the Riemann tensor in the Lagrangian and d corresponds to the number of dimensions. This allows to obtain several results. For instance, the critical compressibility factor Z is a function of n and d that satisfies Z<1 strictly, matching the behaviour of a real gas, but the new values computed differ from the 3/8 value of a Van der Waals gas except for d=4 and n=1. New versions of the Smarr formula and equation of state and its behavior near the critical points are computed, which are also functions of n, d and Z. For all the cases the critical exponent are similar to those of the Van der Waals fluid. The first law of thermodynamics, in the extended space, is deduced by the variation of parameters of the Pure Lovelock solution. The entropy, volume and electric potential are consistent with the previously known results in the literature.