The modified first laws of thermodynamics of anti-de Sitter and de Sitter space–times

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We modify the first laws of thermodynamics of a Reissner–Nordstrom anti-de Sitter black hole and a pure de Sitter space–time by the surface tensions. The corresponding Smarr relations are obeyed. The cosmological constants are first treated as fixed constants, and then as variables associated to the pressures. For the black hole, the law is written as δE=TδS−σδA when the cosmological constant is fixed, where E is the Misner–Sharp mass and σ is the surface tension. Adopting the varied constant, we modify the law as δE0=TδS−σeffδA+VδP, where E0=M−Q22r+ is the enthalpy. The thermodynamical properties are investigated. For the de Sitter space–time, the expressions of the modified laws are different from these of the black hole. The differential way to derive the law is discussed.