  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. Here, we study the mathematical
structure of the physical process first law for a general diffeomorphism
invariant theory of gravity. We analyze the effect of ambiguities in the Wald's
definition of entropy on the physical process first law. We show that for
linearized perturbations, the integrated version of the physical process law,
which determines the change of entropy between two asymptotic cross-sections,
is independent of these ambiguities. In case of entropy change between two
intermediate cross sections of the horizon, we show that it inherits additional
contributions, which 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 for both general relativity and Lanczos-Lovelock
gravity.
