Holographic Black Hole Chemistry

Thermodynamic quantities associated with black holes in Anti-de Sitter space obey an interesting identity when the cosmological constant is included as one of the dynamical variables, the generalized Smarr relation. We show that this relation can easily be understood from the point of view of the dual holographic field theory. It amounts to the simple statement that the extensive thermodynamic quantities of a large $N$ gauge theory only depend on the number of colors, $N$, via an overall factor of $N^2$.