© Copyright owned by the author(s). The role of the cosmological constant as a thermodynamic variable in black hole thermodynamics is discussed. A black hole in asymptotically anti-de Sitter space-time has a positive thermodynamic pressure associated with the cosmological constant and the conjugate thermodynamic variable is a volume. Isotherms in the P − V plane can be constructed and there can be phase transitions between different black hole phases with critical points. Applications to the thermodynamics of conformal field theories are discussed, via the AdS/CFT correspondence. In particular critical exponents are calculated for the deconfining phase transition in N = supersymmetric SU (N) Yang-Mills theory, in the limit of large N, and found to be mean field in this limit.