Does p-isomorphism preserve phase transition? Consider two NP-complete languages that are polynomial-time isomorphic. If we know that one of them exhibits phase transition (with respect to some order parameter), does this imply that the other must also have phase transition, with respect to some (possibly different) order parameter?

Intuitively, I would expect that the answer is yes. The reason is that p-isomorphmism means that the two languages are "essentially the same," apart from a polynomial-time re-encoding. It would appear quite surprising if one of them has phase transition, while the other does not, despite their p-isomorphism. But how can the implication be proved formally?
Yes, but I'm not sure it means much. Yes in a trivial way: suppose $\varphi$ is an isomorphism between two $\mathsf{NP}$-complete languages $L_1, L_2$, and $L_1$ exhibits a phase transition with respect to some parameter $m(x)$. Then so does $L_2$, with respect to the parameter $m_2(y) := m(\varphi^{-1}(y))$. (This relies on the fact that "phase transition" talks about the fraction of satisfiable instances jumping from 0 to 1, as a fraction of an exponential number of instances of a given size, and p-isomorphisms preserve densities up to a polynomial.)

I'm not sure this is such a meaningful statement, however, because the parameter $m_2$ above can be totally unnatural from the point of view of $L_2$. Most phase transitions of interest are with respect to some natural parameter (e.g. clause-to-variable ratio in k-SAT, density of a graph, etc.). But those natural parameters are defined in terms of some "internal" structure to the instances, not by just treating the problem instances as abstract strings in $\Sigma^*$ ($\Sigma$ your finite alphabet). But p-isomorphism does just treat inputs as abstract strings, and "knows nothing" about the internal structure that we like to impose on them. So, sure, you can say that all paddable languages have a phase transition since k-SAT does. But that doesn't tell you if, for example, solvability of quadratic Diophantine equations exhibits a phase transition in terms of any parameter that is natural from the point of view of Diophantine equations...