  For a compact hyperbolic surface, we define a smooth linear statistic,
mimicking the number of Laplace eigenvalues in a short energy window. We study
the variance of this statistic, when averaged over the moduli space $\mathcal
M_g$ of all genus $g$ surfaces with respect to the Weil-Petersson measure. We
show that in the double limit, first taking the large genus limit and then the
high energy limit, we recover GOE statistics. The proof makes essential use of
Mirzakhani's integration formula.
