The cosmological constant and the black hole equation of state

TL;DR

The paper reframes black-hole thermodynamics by treating the cosmological constant as a thermodynamic pressure, with the black-hole mass identified as enthalpy. It derives the enthalpy, temperature, volume, and heat capacities across dimensions, showing that the Gibbs free energy governs the Euclidean action and that a Smarr-like relation $M = 2(TS - PV)$ holds, implying a Hawking-Page transition under appropriate curvature and dimensional conditions. The analysis extends to higher dimensions and various horizon geometries, and provides a detailed BTZ case with quantum corrections that modify the thermodynamic volume. The framework has potential implications for AdS/CFT and condensed-matter applications where constant-pressure thermodynamics and specific heats at constant pressure are particularly relevant.

Abstract

The thermodynamics of black holes in various dimensions are described in the presence of a negative cosmological constant which is treated as a thermodynamic variable, interpreted as a pressure in the equation of state. The black hole mass is then identified with the enthalpy, rather than the internal energy, and heat capacities are calculated at constant pressure not at constant volume. The Euclidean action is associated with a bridge equation for the Gibbs free energy and not the Helmholtz free energy. Quantum corrections to the enthalpy and the equation of state of the BTZ black hole are studied.