  We prove extreme value laws for cusp excursions of the horocycle flow in the
case of surfaces of constant negative curvature. The key idea of our approach
is to study the hitting time distribution for shrinking Poincar\'e sections
that have a particularly simple scaling property under the action of the
geodesic flow. This extends the extreme value law of Kirsebom and Mallahi-Karai
[arXiv:2209.07283] for cusp excursions for the modular surface. Here we show
that the limit law can be expressed in terms of Hall's formula for the gap
distribution of the Farey sequence.
