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We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose densities are regular and vary differentiably with respect to the dynamical system. The maps we consider are skew-products given by 
$T(x,y) = (E (x), C(x,y))$, where E is an expanding map of 
$\mathbb {T}^u$ and C is a contracting map on each fiber. If 
$\inf |\!\det DT| \inf \| (D_yC)^{-1}\| ^{-2s}>1$ for some 
${s<r-(({u+d})/{2}+1)}$, 
$r \geq 2$, and T satisfies a transversality condition between overlaps of iterates of T (a condition which we prove to be 
$C^r$-generic under mild assumptions), then the SRB measure 
$\mu _T$ of T is absolutely continuous and its density 
$h_T$ belongs to the Sobolev space 
$H^s({\mathbb {T}}^u\times {\mathbb {R}}^d)$. When 
$s> {u}/{2}$, it is also valid that the density 
$h_T$ is differentiable with respect to T. Similar results are proved for thermodynamical quantities for potentials close to the geometric potential.
