We study large deviations for measurable averaging operators on state spaces of dynamical systems. Our main motivation is the Hecke operators on the modular curve Y_0(p^n) and their generalization to higher rank S-arithmetic quotients. We prove a relatively sharp large deviations result in terms of the norm of the averaging operator restricted to the orthogonal complement of the constant functions in L2. In the self-adjoint case this norm is expressible by the spectral gap. Developing ideas of Linnik and Ellenberg, Michel and Venkatesh, we use this large deviation result to prove an effective equidistribution theorem on a state space. The novelty of our results is that they apply to measures with sub-optimal bounds on the mass of Bowen balls. We present two new applications to our effective equidistribution result. The first one is effective rigidity for the measure of maximal entropy on S-arithmetic quotients with respect to a semisimple action in a non-archimedean place. Measures having large enough metric entropy must also be close on the state space to the Haar measure. This is a partial extension of a recent result of Rühr to a significantly more general setting. The second one is non-escape of mass for sequences of measures having large entropy with respect to a semisimple element in a non-archimedean place. This generalizes similar known results for real flows. Our methods differ from the methods used by Rühr and in the previously known non-escape of mass results.