  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
behaviour 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. 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 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.
