  This paper provides a rigorous study of the localization transition for a
Gaussian free field on $\mathbb{Z}^d$ interacting with a quenched disordered
substrate that acts on the interface when the interface height is close to
zero. The substrate has the tendency to localize or repel the interface at
different sites and one can show that a localization-delocalization transition
takes place when varying the average pinning potential $h$: the free energy
density is zero in the delocalized regime, that is for $h$ smaller than a
threshold $h_c$, and it is positive for $h>h_c$. For $d\ge 3$ we compute $h_c$
and we show that the transition happens at the same value as for the annealed
model. However we can show that the critical behavior of the quenched model
differs from the one of the annealed one. While the phase transition of the
annealed model is of first order, we show that the quenched free energy is
bounded above by $ (h-h_c)_+^2$ times a positive constant and that, for
Gaussian disorder, the quadratic behavior is sharp. Therefore this provides an
example in which a {\sl relevant disorder critical exponent} can be made
explicit: in theoretical physics disorder is said to be {\sl relevant} when the
disorder changes the critical behavior of a system and, while there are cases
in which it is known that disorder is relevant, the exact critical behavior is
typically unknown. For $d=2$ we are not able to decide whether the quenched and
annealed critical points coincide, but we provide an upper bound for the
difference between them.
