Generating Kerr–anti–de Sitter thermodynamics

In the present work we study the construction of different thermodynamic descriptions for the Kerr-anti–de Sitter (KadS) black holes. The early versions of the KadS thermodynamics are briefly discussed, highlighting some of its strong points and shortcomings. Isohomogeneous transformations, a procedure for generating new thermodynamics, are presented and geometrically interpreted for KadS. This tool is used to determine possible KadS thermodynamics that can be constructed to satisfy a Smarr formula, and the validity of the first law in the generated thermodynamics. The connection between new thermodynamic theories and early Hawking's approach is considered. In this new framework, the usual KadS thermodynamics is complemented with its geometric construction, and Hawking's proposal, which does not satisfy the first law, is improved to an alternative thermodynamic theory. With the quantum statistical relation, Hawking's and this alternative KadS thermodynamics are also generalized from four to higher dimensions.