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Aql X-1 is a prolific transient neutron star low-mass X-ray binary thatexhibits an accretion outburst approximately once every year. Whether thethermal X-rays detected in intervening quiescent episodes are the result ofcooling of the neutron star or due to continued low-level accretion remainsunclear. In this work we use Swift data obtained after the long and bright 2011and 2013 outbursts, as well as the short and faint 2015 outburst, toinvestigate the hypothesis that cooling of the accretion-heated neutron starcrust dominates the quiescent thermal emission in Aql X-1. We demonstrate thatthe X-ray light curves and measured neutron star surface temperatures areconsistent with the expectations of the crust cooling paradigm. By using athermal evolution code, we find that ~1.2-3.2 MeV/nucleon of shallow heatrelease describes the observational data well, depending on the assumedmass-accretion rate and temperature of the stellar core. We find no evidencefor varying strengths of this shallow heating after different outbursts, butthis could be due to limitations of the data. We argue that monitoring Aql X-1for up to ~1 year after future outbursts can be a powerful tool to break modeldegeneracies and solve open questions about the magnitude, depth and origin ofshallow heating in neutron star crusts.

Constraining the properties of neutron star crusts with the transient low-mass X-ray binary Aql X-1