  We investigate the second law of black hole mechanics in gravitational
theories with higher derivative terms in the action. Wall has described a
method for defining an entropy that satisfies the second law to linear order in
perturbations around a stationary black hole. We show that this can be extended
to define an entropy that satisfies the second law to quadratic order in
perturbations, provided that one treats the higher derivative terms in the
sense of effective field theory. We also address some outstanding issues with
Wall's method, in particular, its gauge invariance and its relation to the
Iyer-Wald entropy.
