Is it possible to infer on the thermodynamics of two problems if a reduction from $B$ to $A$ exists?
With thermodynamics you have to be careful with the kind of reductions you allow, or (as Peter Shor pointed out) there can be essentially no thermodynamic relationship implied by a reduction. For example, if we consider not just the complexity of languages, but also the complexity of functions, every language is equivalent (under pretty simple reductions) to a permutation of $\Sigma^*$, and permutations can in principle be computed with zero thermodynamic cost.