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In the resource theory of thermodynamics, the decrease of the free energybased on von Neumann entropy is not a sufficient condition to determine freeevolution. Rather, a whole family of generalised free energies $F_{\alpha}$must be monotonically decreasing. We study the resilience of this result torelaxations of the framework. We use a toy collisional model, in which thedeviations from the ideal situation can be described as arising frominhomogeneities of local fields or temperatures. For any small amount ofperturbation, we find that there exist initial states such that bothsingle-shot and averaged values of $F_{\alpha}$ do not decrease monotonicallyfor all $\alpha>0$. A geometric representation accounts for the observedbehavior in a graphic way.

Violation of all the second laws of thermal operations by inhomogeneous reservoirs