<p>We consider random <i>n</i>-covers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(X_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> of an arbitrary compact hyperbolic surface <i>X</i>. We show that in the large <i>n</i> regime and small window limit, the variance of the smooth spectral statistics of the Laplacian <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Delta _\rho \)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>ρ</mi> </msub> </math></EquationSource> </InlineEquation> twisted by a unitary representation, obey the universal laws of GOE and GUE random matrices, depending on wether the representation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\rho \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> preserves or breaks the time reversal symmetry. These results are in accordance with the semiclassical heuristics of Berry (Stochastic processes in classical and quantum systems, Springer, Berlin, 1986; Chaotic behavior of deterministic systems, North-Holland, Amsterdam, 1983) and are a discrete analog of a recent work of Rudnick (Geom. Funct. Anal. 33(6), 1581-1607 2023) for the Weil-Petersson model of random surfaces.</p>