We consider diffeomorphism invariant theories of gravity with arbitrary
higher derivative terms in the Lagrangian as corrections to the leading two
derivative theory of Einstein’s general relativity. We construct a proof of the
zeroth law of black hole thermodynamics in such theories. We assume that a
stationary black hole solution in an arbitrary higher derivative theory can be
obtained by starting with the corresponding stationary solution in general
relativity and correcting it order by order in a perturbative expansion in the
coupling constants of the higher derivative Lagrangian. We prove that surface
gravity remains constant on its horizon when computed for such stationary black
holes, which is the zeroth law. We argue that the constancy of surface gravity
on the horizon is related to specific components of the equations of motion in
such theories. We further use a specific boost symmetry of the near horizon
space-time of the stationary black hole to constrain the off-shell structure of
the equations of motion. Our proof for the zeroth law is valid up to arbitrary
order in the expansion in the higher derivative couplings.
