  Elastic interfaces display scale-invariant geometrical fluctuations at
sufficiently large lengthscales. Their asymptotic static roughness then follows
a power-law behavior, whose associated exponent provides a robust signature of
the universality class to which they belong. The associated prefactor has
instead a non-universal amplitude fixed by the microscopic interplay between
thermal fluctuations and disorder, usually hidden below experimental
resolution. Here we compute numerically the roughness of a one-dimensional
elastic interface subject to both thermal fluctuations and a quenched disorder
with a finite correlation length. We evidence the existence of a novel
power-law regime at short lengthscales. We determine the corresponding exponent
$\zeta_\textrm{dis}$ and find compelling numerical evidence that, contrarily to
available analytic predictions, one has $\zeta_\textrm{dis} < 1$. We discuss
the consequences on the temperature dependence of the roughness and the
connection with the asymptotic random-manifold regime at large lengthscales. We
also discuss the implications of our findings for other systems such as the
Kardar-Parisi-Zhang equation and the Burgers turbulence.
