  In this paper we consider an exactly solvable model which displays glassy
behavior at zero temperature due to entropic barriers. The new ingredient of
the model is the existence of different energy scales or modes associated to
different relaxational time-scales. Low-temperature relaxation takes place by
partial equilibration of successive lower energy modes. An adiabatic scaling
solution, defined in terms of a threshold energy scale $\eps^*$, is proposed.
For such a solution, modes with energy $\eps\gg\eps^*$ are equilibrated at the
bath temperature, modes with $\eps\ll\eps^*$ remain out of equilibrium and
relaxation occurs in the neighborhood of the threshold $\eps\sim \eps^*$. The
model is presented as a toy example to investigate conditions related to the
existence of an effective temperature in glassy systems and its possible
dependence on the energy sector probed by the corresponding observable.
