  The gravitino problem is investigated in the framework of Extended Gravity
cosmologies. In particular, we consider $f(R)$ gravity, the most natural
extension of the Hilbert-Einstein action, and $f(\cal T)$ gravity, the
extension of teleparallel equivalent gravity. Since in these theories the
expansion laws of the Universe are modified, as compared to the standard
$\Lambda$CDM cosmology, it follows that also the thermal history of particles
gets modified. We show that $f(R)$ models allow to avoid the late abundance of
gravitinos. In particular, we found that for an appropriate choice of the
parameters characterizing the $f(R)$ model, the gravitino abundance turns out
to be independent of the reheating temperature. A similar behavior is achieved
also in the context of $f(\cal T)$ gravity. In this perspective, we can
conclude that geometric corrections to standard General Relativity (and to
Teleparallel Equivalent of General Relativity) can improve shortcomings both in
cosmology and in unified theories beyond the Standard Model of particles.
