Peter Shor commented on this post:
years of experience in theoretical computer science says that the thermodynamic behavior of two NP complete problems are in general not similar.
What can we say about the distribution/thermodynamics of $A$ w.r.t. $B$ if we know that $A \le_M B$ (there is a reduction from $B$ to $A$), or if we know that $A, B \in C$ where $C$ is some complexity class ($P$, $NP$, co-$NP$, etc.)?
What can we say on the thermodynamics if there isn't a reduction from $A$ to $B$, or given their complexity classes?
I am not sure if it is a wicked idea to use complexity theory whose main concern is the worst-case scenarios while discussing thermodynamics, or statistical physics, whose general aim is to characterize phase transitions and describe the average case (principle of maximum entropy). Here is somehow a related work that states "NP problems" in terms of an Ising model.