The entropy of a dynamical black hole equals the area of its apparent horizon at second order in perturbations when the null energy condition holds.
In the following, we prove this statement using the method of [29], together with the boundary conditions and a consequence of the null energy condition
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The entropy of black hole under second-order deviation from equilibrium
The entropy of a dynamical black hole equals the area of its apparent horizon at second order in perturbations when the null energy condition holds.