  We study the stability of statistical properties of Anosov maps on tori by
examining the stability of the spectrum of an analytically twisted
Perron-Frobenius operator on the anisotropic Banach spaces of Gou\"ezel and
Liverani. By extending our previous work in [8], we obtain the stability of
various statistical properties (the variance of a CLT and the rate function of
an LDP) of Anosov maps to general perturbations, including new classes of
numerical approximations. In particular, we obtain new results on the stability
of the rate function under deterministic perturbations. As a key application,
we focus on perturbations arising from numerical schemes and develop two new
Fourier-analytic method for efficiently computing approximations of the
aforementioned statistical properties. This includes the first example of a
rigorous scheme for approximating the peripheral spectral data of the
Perron-Frobenius operator of an Anosov map without mollification. Using the two
schemes we obtain the first rigorous estimates of the variance and rate
function for Anosov maps.
