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Traditional treatments of the first law of black hole thermodynamics do notinclude a discussion of pressure and volume. We give an overview of recentdevelopments proposing a definition of volume that can be used to extend thefirst law to include these appropriately. New results are also presentingrelating to the critical point and the associated second order phase transitionfor a rotating black-hole in four-dimensional space-time which isasymptotically anti-de Sitter. In line with known results for a non-rotatingcharged black-hole, this phase transition is shown to be of Van der Waals typewith mean field exponents.

Where is the PdV term in the first law of black hole thermodynamics?