The evolution of antimicrobial resistance generally occurs in an environment where antimicrobial concentration is variable, which has dramatic consequences on the microorganisms' fitness landscape, and thus on the evolution of resistance. We investigate the effect of these time-varying patterns of selection within a stochastic model. We consider a homogeneous microbial population of fixed size subjected to periodic alternations of phases of absence and presence of an antimicrobial that stops growth. Combining analytical approaches and stochastic simulations, we quantify how the time necessary for fit resistant bacteria to take over the microbial population depends on the alternation period. We demonstrate that fast alternations strongly accelerate the evolution of resistance, reaching a plateau for sufficiently small periods. Furthermore, this acceleration is stronger in larger populations. For asymmetric alternations, featuring a different duration of the phases with and without antimicrobial, we shed light on the existence of a minimum for the time taken by the population to fully evolve resistance. The corresponding dramatic acceleration of the evolution of antimicrobial resistance likely occurs in realistic situations, and may have an important impact both in clinical and experimental situations. (C) 2018 Elsevier Ltd. All rights reserved.