  We consider a $\mathbb R/\mathbb Z$ extension of an Anosov diffemorphism of a
compact Riemannian manifold by a random function $\tau$ and show that the flat
traces of the transfer operator, reduced with respect to frequency in the
fibers, converge in law towards Gaussians, up to an Ehrenfest time that
decreases with the regularity of $\tau$.
