The time-integrated intensity transmitted by a laser-driven resonator obeys Lévy's arcsine laws [Ramesh , ]. Here, we demonstrate the implications of these laws for optical sensing. We consider the standard goal of resonant optical sensors, namely to report a perturbation to their resonance frequency. In this context, we quantify the sensing precision attained using a finite-energy budget combined with time or ensemble averaging of the time-integrated intensity. We find that ensemble averaging outperforms time averaging for short observation times, but the advantage disappears as the observation time increases. We explain this behavior in terms of weak ergodicity breaking, which arises when the time for the time-integrated intensity to explore the entire phase space diverges, while the observation time remains finite. Evidence that the former time diverges is presented in first passage and return time distributions. Our results are relevant to all types of sensors, in optics and beyond, where stochastic time-integrated fields or intensities are measured to detect an event. In particular, choosing the right averaging strategy can improve sensing precision by orders of magnitude with zero-energy cost. Published by the American Physical Society 2024