  We consider random n-covers $X_n$ of an arbitrary compact hyperbolic surface
$X$. We show that in the large n regime and small window limit, the variance of
the smooth spectral statistics of the Laplacian twisted by a unitary abelian
character, obey the universal laws of GOE and GUE random matrices, depending on
wether the character preserves or breaks the time reversal symmetry. We also
prove a generalization for higher dimensional twists valued in compact linear
groups. These results confirm a conjecture of Berry and is a discrete analog of
a recent work of Rudnick for the Weil-Petersson model of random surfaces.
