  We construct a proof of the second law of thermodynamics in an arbitrary
diffeomorphism invariant theory of gravity working within the approximation of
linearized dynamical fluctuations around stationary black holes. We achieve
this by establishing the existence of an entropy current defined on the horizon
of the dynamically perturbed black hole in such theories. By construction, this
entropy current has non-negative divergence, suggestive of a mechanism for the
dynamical black hole to approach a final equilibrium configuration via entropy
production as well as the spatial flow of it on the null horizon. This enables
us to argue for the second law in its strongest possible form, which has a
manifest locality at each space-time point. We explicitly check that the form
of the entropy current that we construct in this paper exactly matches with
previously reported expressions computed considering specific four derivative
theories of higher curvature gravity. Using the same set up we also provide an
alternative proof of the physical process version of the first law applicable
to arbitrary higher derivative theories of gravity.
