  With our neutron star crust cooling code {\tt NSCool} we track the thermal
evolution of the neutron star in Aql X-1 over the full accretion outburst
history from 1996 until 2015. For the first time, we model many outbursts (23
outbursts were detected) collectively and in great detail. This allows us to
investigate the influence of previous outbursts on the internal temperature
evolution and to test different neutron star crust cooling scenarios. Aql X-1
is an ideal test source for this purpose, because it shows frequent, short
outbursts and thermally dominated quiescence spectra. The source goes into
outburst roughly once a year for a few months.
  Assuming that the quiescent {\it Swift}/XRT observations of Aql X-1 can be
explained within the crust cooling scenario (Waterhouse et al. 2016), we find
three main conclusions. Firstly, the data are well reproduced by our model if
the envelope composition and shallow heating parameters are allowed to change
between outbursts. This is not the case if both shallow heating parameters
(strength and depth) are tied throughout all accretion episodes, supporting
earlier results that the properties of the shallow heating mechanism are not
constant between outbursts. Second, from our models shallow heating could not
be connected to one specific spectral state during outburst. Third, and most
importantly, we find that the neutron star in Aql X-1 does not have enough time
between outbursts to cool down to crust-core equilibrium and that heating
during one outburst influences the cooling curves of the next.
