  We perform a large deviations analysis of homological growth rates of
oriented geodesics on hyperbolic surfaces. For surfaces uniformized by a wide
class of Fuchsian groups of the first kind, we prove the existence of the rate
function which estimates exponential probabilities with which the homological
growth rates stay away from the mean value. The rate function is given in terms
of the multifractal dimension spectrum described in our earlier result
[arXiv:2204.08907]. We also establish an Erd\H{o}s-R\'enyi law, and refined
large deviations upper bounds.
