Abstract: Hartshorne-Speiser-Lyubeznik numbers (HSL numbers) are a measure of nilpotency of the action of Frobenius on the local cohomology modules of a commutative Noetherian ring of positive characteristic. By the Hartshorne-Speiser-Lyubeznik theorem these HSL numbers are finite, but in general an explicit upper bound for these numbers is not known. In this talk we present an explicit upper bound for the HSL numbers of toric face rings and semigroup rings, and discuss the differences in these two bounds.