  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 dE=TdS-PdV, where P is
identified with the total pressure of all matter in the spacetime (including a
cosmological constant Lambda 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.
