Holographic heat engines and static black holes: a general efficiency formula

We study the efficiency of holographic heat engines in the context of extended black hole thermodynamics, where the cosmological constant becomes a dynamical variable. By taking the working substance as a static black hole (i.e. a space-time with vanishing $C_V$) we derive an exact and analytic efficiency formula for virtually any engine defined by a cycle in the $p-V$ plane. This formula gives a simple criteria which completely resolves the benchmarking program for static black holes and shows that for any given engine there is an infinite family of tractable deformations which leave the efficiency invariant. We also derive an upper bound for the efficiency that holds for general engines.