  We consider a class of endomorphisms that contains a set of piecewise
partially hyperbolic dynamics semi-conjugated to non-uniformly expanding maps.
Our goal is to study a class of endomorphisms that preserve a foliation that is
almost everywhere uniformly contracted, with possible discontinuity sets
parallel to the contracting direction. We apply the spectral gap property and
the $\zeta$-H\"older regularity of the disintegration of its equilibrium states
to prove a quantitative statistical stability statement. More precisely, under
deterministic perturbations of the system of size $\delta$, we show that the
$F$-invariant measure varies continuously with respect to a suitable
anisotropic norm. Moreover, we prove that for certain interesting classes of
perturbations, its modulus of continuity is $O(\delta^\zeta \log \delta)$. This
article has been accepted for publication in the Discrete and Continuous
Dynamical Systems journal.
