Is it there any relation between an action functional and entropy? I've found papers that seem to suggest that these concepts are the same, like this one: https://arxiv.org/abs/1005.3854
But I've found answers in Physics Stack Exchange that say that both are incompatible. Although other answers say they are related: Entropy and the principle of least action 
I hope you could clarify this confusion. Why is this so unclear? Why are there contradicting arguments?
In summary: When there is an action, does it have any associated/related entropy? Is every action related to entropy? Are these the same? Are these completely incompatible?? Has physics advanced enough to know a definitive answer?
 A: There is no universal relation in standard accepted physics. Nowadays there are attempts to find conceptual link between these and also other attractive, so far unconnected concepts, but one must be wary of surface similarities and cheap analogies.
Entropy in its original meaning is a characteristic of state beyond the scope of pure mechanics. Later, different kinds of entropy were introduced, for example entropy that is a function of probability distribution on some space.
Action, on the other hand, is a characteristic of a process in time and space in mechanics or in field theory. It does not depend on any probability, but depends on a trajectory.
These are concepts from different theories describing different kinds of things.
A: We have shown that entropy is a logarithmic function of the action ratio (@/hbar) and the entropy of atmospheric gases is easily calculated using this approach (Kennedy et al. Entropy, 21, 45, 2019). Entropy indicates the scale of energy needed to sustain the action of a molecular system. This approach provides a new way to understand the Carnot cycle (Entropy 23, 860, 2021 and to understand chemical reaction rates.  At event horizons, the bits of quantum action contained in the horizon are a direct measure of the entropy.
