  We consider families of random products of close-by Anosov diffeomorphisms,
and show that statistical stability and linear response hold for the associated
families of equivariant and stationary measures. Our analysis rely on the study
of the top Oseledets space of a parametrized transfer operator cocycle, as well
as ad-hoc abstract perturbation statements. As an application, we show that,
when the quenched central limit theorem holds, under the conditions that ensure
linear response for our cocycle, the variance in the CLT depends differentiably
on the parameter.
