I agree that simulated annealing is perhaps the closest. The missing link, I suppose, is knowing what distribution you get by running the algorithm for a fixed temperature (alpha). But the method works even if you don't know all that really matters is that it converges to the distribution you want, which is neat! I might use that trick sometime.
The reason simulated annealing sprang to mind is that I've done something similar once where I tuned the temperature curve for a simulated annealing algorithm, but I couldn't find a general enough rule. So seeing someone do something similar is really neat!
There are some theoretical guarantees to be had if the update step can be recast into a MCMC update, or equivalent. And I think there's an expression for the optimal temperature at any point in time (same way you found it for alpha). Any idea if there is a useful description for the steady state distribution at a given alpha?