Let's say I'm trying to sample from a posterior distribution, $p(\theta|x)$, which I suspect is very "bumpy" (10s to 1000s of small modes). Moreover, the evaluation of the posterior kernel $K(\theta|x) \propto p(\theta|x)$ is very expensive to evaluate (and not conjugate) and the data set is large enough such that I would like to avoid any technique based on parallel chains.
However, I would like to avoid being stuck in any one mode for too long during the sampling process, so I employ a standard annealed MCMC technique (such as Metroplis-Hastings with an annealed acceptance probability) but I never cool the temperature parameter. The distribution I'm sampling from is thus "flattened", and easier to sample from in the case of many modes, but the stationary distribution for the Markov chain has clearly changed from the one defined by $K(\theta|x)$.
Does anyone have any references about what happens to the stationary distribution of the Markov chain in this case?