Abstract: The Hartshorne-Speiser-Lyubeznik number (HSL number) is a nilpotency index for modules with a Frobenius action. One important class of such modules is the local cohomology modules of a ring of positive characteristic. The HSL numbers of these local cohomology modules are closely connected to the class of F-nilpotent singularities, which is of interest in part due to its connection to other F-singularities. In this talk we present an explicit upper bound for the HSL numbers of the local cohomology modules of a positive characteristic pointed semigroup ring.