  We start by presenting a brief summary of fractional quantum mechanics, as
means to convey a motivation towards fractional quantum cosmology.
Subsequently, such application is made concrete with the assistance of a case
study. Specifically, we investigate and then discuss a model of stiff matter in
a spatially flat homogeneous and isotropic universe. A new quantum cosmological
solution, where fractional calculus implications are explicit, is presented and
then contrasted with the corresponding standard quantum cosmology setting.
