Treating the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we reconsider the critical behaviour of charged AdS black holes. We complete the analogy of this system with the liquid-gas system and study its critical point, which occurs at the point of divergence of specific heat at constant pressure. We calculate the critical exponents and show that they coincide with those of the Van der Waals system.

In a theory where the cosmological constant Λ or the gauge coupling constant g arises as the vacuum expectation value, its variation should be included in the first law of thermodynamics for black holes. This becomes dE = T dSwhere E is now the enthalpy of the spacetime, and Θ, the thermodynamic conjugate of Λ, is proportional to an effective volume V = − 16πΘ D−2 "inside the event horizon." Here we calculate Θ and V for a wide variety of D-dimensional charged rotating asymptotically AdS black hole spacetimes, using the first law or the Smarr relation. We compare our expressions with those obtained by implementing a suggestion of Kastor, Ray and Traschen, involving Komar integrals and Killing potentials, which we construct from 1 conformal Killing-Yano tensors. We conjecture that the volume V and the horizon areaA D−2 is the volume of the unit (D − 2)-sphere, and we show that this is obeyed for a wide variety of black holes, and saturated for Schwarzschild-AdS. Intriguingly, this inequality is the "inverse" of the isoperimetric inequality for a volume V in Euclidean (D−1) space bounded by a surface of area A, for which R ≤ 1. Our conjectured Reverse Isoperimetric Inequality can be interpreted as the statement that the entropy inside a horizon of a given "volume" V is maximised for Schwarzschild-AdS. The thermodynamic definition of V requires a cosmological constant (or gauge coupling constant).However, except in 7 dimensions, a smooth limit exists where Λ or g goes to zero, providing a definition of V even for asymptotically-flat black holes.2

We investigate the critical behaviour of charged and rotating AdS black holes in d spacetime dimensions, including effects from non-linear electrodynamics via the Born-Infeld action, in an extended phase space in which the cosmological constant is interpreted as thermodynamic pressure. For Reissner-Nördstrom black holes we find that the analogy with the Van der Walls liquid-gas system holds in any dimension greater than three, and that the critical exponents coincide with those of the Van der Waals system. We find that neutral slowly rotating black holes in four spacetime dimensions also have the same qualitative behaviour. However charged and rotating black holes in three spacetime dimensions do not exhibit critical phenomena. For Born-Infeld black holes we define a new thermodynamic quantity B conjugate to the Born-Infeld parameter b that we call Born-Infeld vacuum polarization. We demonstrate that this quantity is required for consistency of both the first law of thermodynamics and the corresponding Smarr relation.

We review recent developments on the thermodynamics of black holes in extended phase space, where the cosmological constant is interpreted as thermodynamic pressure and treated as a thermodynamic variable in its own right. In this approach, the mass of the black hole is no longer regarded as internal energy, rather it is identified with the chemical enthalpy. This leads to an extended dictionary for black hole thermodynamic quantities, in particular a notion of thermodynamic volume emerges for a given black hole spacetime. This volume is conjectured to satisfy the reverse isoperimetric inequalityan inequality imposing a bound on the amount of entropy black hole can carry for a fixed thermodynamic volume. New thermodynamic phase transitions naturally emerge from these identifications. Namely, we show that black holes can be understood from the viewpoint of chemistry, in terms of concepts such as Van der Waals fluids, reentrant phase transitions, and triple points. We also review the recent attempts at extending the AdS/CFT dictionary in this setting, discuss the connections with horizon thermodynamics, applications to Lifshitz spacetimes, and outline possible future directions in this field.