Higher Order Corrections to Holographic Black Hole Chemistry

We investigate the holographic Smarr relation beyond the large N limit. By making use of the holographic dictionary, we find that the bulk correlates of sub-leading 1/N corrections to this relation are related to the couplings in Lovelock gravity theories. We likewise obtain a holographic equation of state, and check its validity for a variety of interesting and non-trivial black holes, including rotating planar black holes in Gauss-Bonnet-Born-Infeld gravity, and non-extremal rotating black holes in minimal 5d gauged supergravity. We provide an explanation of the N-dependence of the holographic Smarr relation in terms of contributions due to planar and non-planar diagrams in the dual theory.