  We study the convergence time to equilibrium of the Metropolis dynamics for
the Generalized Random Energy Model with an arbitrary number of hierarchical
levels, a finite and reversible continuous-time Markov process, in terms of the
spectral gap of its transition probability matrix. This is done by deducing
bounds to the inverse of the gap using a Poincar\'e inequality and a path
technique. We also apply convex analysis tools to give the bounds in the most
general case of the model.
