Thermodynamics of $\mathrm{}(3+1)\mathrm{}$-dimensional black holes with toroidal or higher genus horizons

We examine counterparts of the Reissner-Nordström–anti–de Sitter black hole spacetimes in which the two-sphere has been replaced by a surface Σ of constant negative or zero curvature. When horizons exist, the spacetimes are black holes with an asymptotically locally anti–de Sitter infinity, but the infinity topology differs from that in the asymptotically Minkowski case, and the horizon topology is not $S^2.$ Maximal analytic extensions of the solutions are given. The local Hawking temperature is found. When Σ is closed, we derive the first law of thermodynamics using a Brown-York-type quasilocal energy at a finite boundary, and we identify the entropy as one-quarter of the horizon area, independent of the horizon topology. The heat capacities with constant charge and constant electrostatic potential are shown to be positive definite. With the boundary pushed to infinity, we consider thermodynamical ensembles that fix the renormalized temperature and either the charge or the electrostatic potential at infinity. Both ensembles turn out to be thermodynamically stable, and dominated by a unique classical solution.