  We show that for a general four derivative theory of gravity, only the
holographic entanglement entropy functionals obey the second law at linearized
order in perturbations. We also derive bounds on the higher curvature couplings
in several examples, demanding the validity of the second law for higher order
perturbations. For the five dimensional Gauss-Bonnet theory in the context of
AdS/CFT, the bound arising from black branes coincides with there being no
sound channel instability close to the horizon. Repeating the analysis for
topological black holes, the bound coincides with the tensor channel causality
constraint (which is responsible for the viscosity bound). Furthermore, we show
how to recover the holographic c-theorems in higher curvature theories from
similar considerations based on the Raychaudhuri equation.
