  We study numerically thermal effects at the depinning transition of an
elastic string driven in a two-dimensional uncorrelated disorder potential. The
velocity of the string exactly at the sample critical force is shown to behave
as $V \sim T^\psi$, with $\psi$ the thermal rounding exponent. We show that the
computed value of the thermal rounding exponent, $\psi = 0.15$, is robust and
accounts for the different scaling properties of several observables both in
the steady-state and in the transient relaxation to the steady-state. In
particular, we show the compatibility of the thermal rounding exponent with the
scaling properties of the steady-state structure factor, the universal
short-time dynamics of the transient velocity at the sample critical force, and
the velocity scaling function describing the joint dependence of the
steady-state velocity on the external drive and temperature.
