The thermodynamics of a KerrNewmande Sitter black hole

We compute the conserved quantities of the four-dimensional Kerr–Newman–de Sitter (KNdS) black hole through the use of the counterterm renormalization method, and obtain a generalized Smarr formula for the mass as a function of the entropy, the angular momentum, and the electric charge. The first law of thermodynamics associated to the cosmological horizon of KNdS is also investigated. Using the minimal number of intrinsic boundary counterterms, we consider the quasilocal thermodynamics of an asymptotic de Sitter–Reissner–Nordstrom black hole, and find that the temperature is equal to the product of the surface gravity (divided by 2π) and the Tolman red-shift factor. We also perform a quasilocal stability analysis by computing the determinant of Hessian matrix of the energy with respect to its thermodynamic variables in both the canonical and the grand-canonical ensembles and obtain a complete set of phase diagrams. We then turn to the quasilocal thermodynamics of four-dimensional Kerr–Newman–de Sitter black hole for virtually all possible values of the mass, the rotation, and the charge parameters that leave the quasilocal boundary inside the cosmological event horizon, and perform a quasilocal stability analysis of KNdS black hole.PACS Nos.:04.70.Dy, 04.62.+v, 04.60.–m