  We study smooth linear spectral statistics of twisted Laplacians on random
$n$-covers of a fixed compact hyperbolic surface $X$. We consider two aspects
of such statistics. The first is the fluctuations of such statistics in a small
energy window around a fixed energy level when averaged over the space of all
degree $n$ covers of $X$. The second is the energy variance of a typical
surface.
  In the first case, we show a central limit theorem. Specifically, we show
that the distribution of such fluctuations tends to a Gaussian with variance
given by the corresponding quantity for the Gaussian Orthogonal/Unitary
Ensemble (GOE/GUE). In the second case, we show that the energy variance of a
typical random $n$-cover is that of the GOE/GUE. In both cases, we consider a
double limit where first we let $n$, the covering degree, go to $\infty$ then
let $L\to \infty$ where $1/L$ is the window length.
