In this review we summarize, expand, and set in context 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. We specifically consider the thermodynamics of higher-dimensional rotating asymptotically flat and AdS black holes and black rings in a canonical (fixed angular momentum) ensemble. We plot the associated thermodynamic potential-the Gibbs free energy-and study its behavior to uncover possible thermodynamic phase transitions in these black hole spacetimes. We show that the multiply-rotating Kerr-AdS black holes exhibit a rich set of interesting thermodynamic phenomena analogous to the "every day thermodynamics" of simple substances, such as reentrant phase transitions of multicomponent liquids, multiple first-order solid/liquid/gas phase transitions, and liquid/gas phase transitions of the van der Waals type. Furthermore, the reentrant phase transitions also occur for multiply-spinning asymptotically flat Myers-Perry black holes. These phenomena do not require a variable cosmological constant, though they are more naturally understood in the context of the extended phase space. The thermodynamic volume, a quantity conjugate to the thermodynamic pressure, is studied for AdS black rings and demonstrated to satisfy the reverse isoperimetric inequality; this provides a first example of calculation confirming the Galaxies 2014, 2 90 validity of isoperimetric inequality conjecture for a black hole with non-spherical horizon topology. The equation of state P = P (V, T ) is studied for various black holes both numerically and analytically-in the ultraspinning and slow rotation regimes.

We study the thermodynamics of higher-dimensional singly spinning asymptotically AdS black holes in the canonical (fixed J) ensemble of extended phase space, where the cosmological constant is treated as pressure and the corresponding conjugate quantity is interpreted as thermodynamic volume. Along with the usual small/large black hole phase transition, we find a new phenomenon of reentrant phase transitions for all d ≥ 6 dimensions, in which a monotonic variation of the temperature yields two phase transitions from large to small and back to large black holes. This situation is similar to that seen in multicomponent liquids. Introduction. In view of the AdS/CFT correspondence, phase transitions in asymptotically AdS black holes allow for a dual interpretation in the thermal conformal field theory (CFT) living on the AdS boundary-the principal example being the well known radiation/Schwarzschild-AdS black hole Hawking-Page transition [1] which can be interpreted as a confinement/deconfinement phase transition in the dual quark gluon plasma [2]. Charged [3-6] and rotating [7,8] asymptotically AdS back holes possess an interesting feature-they allow for a first order small-blackhole/large-black-hole phase (SBH/LBH) transition which is in many ways reminiscent of the liquid/gas transition of the Van der Waals fluid. This superficial analogy was recently found more intriguing [9] by considering a thermodynamic analysis in an extended phase space where the cosmological constant is identified with thermodynamic pressure and its variations are included in the first law of black hole thermodynamics. This notion emerges from geometric derivations of the Smarr formula [10] that i) imply the mass of an AdS black hole should be interpreted as the enthalpy of the spacetime and ii) allow for a computation of the conjugate thermodynamic volume. Intensive and extensive quantities are now properly identified [9] and the SBH/LBH transition can be understood as a liquid/gas phase transition by employing Maxwell's equal area law to the P − V diagram. Coexistence lines and critical exponents are then seen to match those of a Van der Waals fluid.