The normalized volume of a Kawamata log terminal (klt) singularity was introduced by Chi Li, who used it to characterize K-semistability of a complex Fano variety X in terms of the singularity of the cone over X. Moreover, for general (complex) klt singularities, a local stability theory has been established via the study of the normalized volume (and its minimization). In this talk, I will present joint work with Yuchen Liu, who introduced a positive characteristic analog of Li's normalized volume (called F-volume) by incorporating the Frobenius map. I will discuss a Frobenius version of Tian's criterion for K-semistability of Fano varieties by relating the F-volume to the Frobenius-alpha invariant. As applications, we compute the limit F-volumes of two-dimensional singularities via reduction modulo p and show that the limit matches Li's normalized volume. Time permitting, I will also mention connections to the F-signature, which underlies the proof of our main results.