Universality of P − V Criticality in Horizon Thermodynamics

We study P − V criticality of black holes in Lovelock gravities in the context of horizon thermodynamics. The corresponding first law of horizon thermodynamics emerges as one of the Einstein–Lovelock equations and assumes the universal (independent of matter content) form δE = TδS − PδV , where P is identified with the total pressure of all matter in the spacetime (including a cosmological constant Λ if present). We compare this approach to recent advances in extended phase space thermodynamics of asymptotically AdS black holes where the ‘standard’ first law of black hole thermodynamics is extended to include a pressure–volume term, where the pressure is entirely due to the (variable) cosmological constant. We show that both approaches are quite different in interpretation. Provided there is sufficient non-linearity in the gravitational sector, we find that horizon thermodynamics admits the same interesting black hole phase behaviour seen in the extended case, such as a Hawking–Page transition, Van der Waals like behaviour, and the presence of a triple point. We also formulate the Smarr formula in horizon thermodynamics and discuss the interpretation of the quantity E appearing in the horizon first law.