Abstract

In black hole thermodynamics a cosmological constant contributes a pressure to the equation of state whose the conjugate variable is a ‘volume’. It is shown that, for a negative cosmological constant, this ‘thermodynamic volume’ is the volume of space excluded by the event horizon, when quantum gravity corrections are ignored. This applies to black holes in any dimension with any event horizon geometry compatible with Einstein’s equations, such as spheres, tori, spaces of constant negative curvature or, in general, any Einstein space with constant Ricci scalar. Quantum corrections to the volume are calculate for the BTZ black hole.