Thermodynamics of topological black holes in $R^{2}$ gravity

TL;DR

This paper investigates topological black hole solutions in quadratic $R^{2}$ gravity, analyzing their thermodynamics, entropy, and dualities, revealing positive entropy and a structured thermodynamic state space.

Contribution

It identifies two classes of solutions in quadratic gravity, computes their thermodynamic properties, and explores the structure and dualities within their thermodynamic state space.

Findings

Entropy is positive for all horizon geometries.

A consistent generalized first law of thermodynamics is formulated.

The thermodynamic state space has a projective structure with dualities.

Abstract

We study topological black hole solutions of the simplest quadratic gravity action and we find that two classes are allowed. The first is asymptotically flat and mimics the Reissner-Nordstr\"om solution, while the second is asymptotically de Sitter or anti-de Sitter. In both classes, the geometry of the horizon can be spherical, toroidal or hyperbolic. We focus in particular on the thermodynamical properties of the asymptotically anti-de Sitter solutions and we compute the entropy and the internal energy with Euclidean methods. We find that the entropy is positive-definite for all horizon geometries and this allows to formulate a consistent generalized first law of black hole thermodynamics, which keeps in account the presence of two arbitrary parameters in the solution. The two-dimensional thermodynamical state space is fully characterized by the underlying scale invariance of the action and it has the structure of a projective space. We find a kind of duality between black holes and other objects with the same entropy in the state space. We briefly discuss the extension of our results to more general quadratic actions.