  The physical process version of the first law can be obtained for bifurcate
Killing horizons with certain assumptions. Especially, one has to restrict to
the situations where the horizon evolution is quasi-stationary, under
perturbations. We revisit the analysis of this assumption considering the
horizon perturbations of Rindler horizon by a spherically symmetric object. We
demonstrate that even if the quasi-stationary assumption holds, the change in
entropy, in four space-time dimensions, diverges when considered between
asymptotic cross-sections. However, these divergences do not appear in higher
dimensions. We also analyze these features in the presence of a positive
cosmological constant. In the process, we prescribe a recipe to establish the
physical process first law in such ill-behaved scenarios.
