  We propose an entropy current for dynamical black holes in a theory with
arbitrary four derivative corrections to Einstein's gravity, linearized around
a stationary black hole. The Einstein-Gauss-Bonnet theory is a special case of
the class of theories that we consider. Within our approximation, our
construction allows us to write down a completely local version of the second
law of black hole thermodynamics, in the presence of the higher derivative
corrections considered here. This ultra-local, stronger form of the second law
is a generalization of a weaker form, applicable to the total entropy,
integrated over a compact `time-slice' of the horizon, a proof of which has
been recently presented in arXiv:1504.08040. We also provide a general
algorithm to construct the entropy current for the four derivative theories,
which may be straightforwardly generalized to arbitrary higher derivative
corrections to Einstein's gravity. This algorithm highlights the possible
ambiguities in defining the entropy current.
