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

Smooth linear eigenvalue statistics on random covers of compact hyperbolic surfaces -- A central limit theorem and almost sure RMT statistics