Sure, there are fruitful parallels. But actually, energy is much 'realer' or more fundamental here than temperature. BTW, those λ_j Lagrangian multipliers can be negative, so don't map onto inverse temperature well. They're more like interaction weights in a Hopfield network.

But there's one setting in which the energy E(x) = Σ_j λ_j H_j(x) really does become a physical (physics-esque) energy: if you sample from the distribution using Hamiltonian Monte Carlo (an MCMC algorithm). Then E(x) is the potential energy of a sample/particle at position x, which is also given a kinetic energy K with the sume E+K held constant as the particle moves about to explore the distribution. It's a cool algorithm but after having used it I'm still not sure how useful it is.