Physical process first law for dynamical black holes and the membrane paradigm

Physical process version of the first law of black hole mechanics relates the change in entropy of a perturbed Killing horizon, between two asymptotic cross sections, to the matter flow into the horizon. In dynamical situations, for two intermediate cross sections of the horizon, it follows that the change of entropy inherits additional contributions. We show that these additional terms essentially coincide with the membrane energy associated with the horizon fluid. Using this interpretation, we write down a physical process first law for entropy change between two arbitrary non stationary cross sections of the horizon. Moreover, the above result goes beyond general relativity and holds for Lanczos-Lovelock gravity as well.