  We study the emergence of a crossover from entropically driven to thermally
activated dynamics in different versions of the `entropic' phase space model
introduced by Barrat and M\'ezard, and previously considered in the zero
temperature limit. We first focus on the low temperature (T < Tg) aging phase
of the original model and show that a short time singularity appears in the
correlation function for Tg/2 < T < Tg, leading to dynamical ultrametricity at
T=Tg. We then consider the finite size version of this model, showing that the
long time dynamics is always thermally activated beyond a size dependent
crossover time scale. We also generalize the model, introducing a threshold
energy so as to mimic a phase space composed of saddles above the threshold,
and minima below. In this case, the crossover time scale becomes much smaller
than the equilibration time, and both kinds of aging dynamics are successively
found, inducing a non trivial aging scaling which does not reduce to the usual
t/tw one.
