Suppose one is performing Gibbs sampling with a Boltzmann distribution (or if you prefer, simulated annealing) at finite temperature. In general we would want to anneal: as the sampler converges to equilibrium, the temperature should be decreased.
If the sampler's rate of convergence (local in time) does not vary, there is no real reason to vary the temperature. On the other hand, if the convergence rate decreases, then the temperature should be decreased as well. This naturally leads to a heuristic that sets the temperature as (proportional to) the convergence rate.
Has this heuristic (or one like it) been explored in the literature? Specific references and/or buzzwords are most welcome.