  Simulated and parallel tempering are families of Markov Chain Monte Carlo
algorithms where a temperature parameter is varied during the simulation to
overcome bottlenecks to convergence due to multimodality.
  In this work we introduce and analyze the convergence for a set of new
tempering distributions which we call \textit{entropy dampening}. For
asymmetric exponential distributions and the mean field Ising model with and
external field simulated tempering is known to converge slowly. We show that
tempering with entropy dampening distributions mixes in polynomial time for
these models.
  Examining slow mixing times of tempering more closely, we show that for the
mean-field 3-state ferromagnetic Potts model, tempering converges slowly
regardless of the temperature schedule chosen. On the other hand, tempering
with entropy dampening distributions converges in polynomial time to
stationarity. Finally we show that the slow mixing can be very expensive
practically. In particular, the mixing time of simulated tempering is an
exponential factor longer than the mixing time at the fixed temperature.
