  We study the thermally assisted relaxation of a directed elastic line in a
two dimensional quenched random potential by solving numerically the
Edwards-Wilkinson equation and the Monte Carlo dynamics of a solid-on-solid
lattice model. We show that the aging dynamics is governed by a growing
correlation length displaying two regimes: an initial thermally dominated
power-law growth which crosses over, at a static temperature-dependent
correlation length $L_T \sim T^3$, to a logarithmic growth consistent with an
algebraic growth of barriers. We present a scaling arguments to deal with the
crossover-induced geometrical and dynamical effects. This analysis allows to
explain why the results of most numerical studies so far have been described
with effective power-laws and also permits to determine the observed anomalous
temperature-dependence of the characteristic growth exponents. We argue that a
similar mechanism should be at work in other disordered systems. We generalize
the Family-Vicsek stationary scaling law to describe the roughness by
incorporating the waiting-time dependence or age of the initial configuration.
The analysis of the two-time linear response and correlation functions shows
that a well-defined effective temperature exists in the power-law regime.
Finally, we discuss the relevance of our results for the slow dynamics of
vortex glasses in High-Tc superconductors.
