We obtained four Chandra /ACIS-S observations beginning two weeks after the end of the November 2000 outburst of the neutron star ( NS ) transient Aql X - 1 . Over the five month span in quiescence , the X-ray spectra are consistent with thermal emission from a NS with a pure hydrogen photosphere and R _ { \infty } =15.9 ^ { +0.8 } _ { -2.9 } ( d/5 kpc ) km at the optically implied X-ray column density . We also detect a hard power-law tail during two of the four observations . The intensity of Aql X - 1 first decreased by 50 \pm 4 % over three months , then increased by 35 \pm 5 % in one month , and then remained constant ( < 6 % change ) over the last month . These variations in the first two observations can not be explained by a change in the power-law spectral component , nor in the X-ray column density . Presuming that R _ { \infty } is not variable and a pure hydrogen atmosphere , the long-term changes can only be explained by variations in the NS effective temperature , from kT _ { eff, \infty } =130 ^ { +3 } _ { -5 } eV , down to 113 ^ { +3 } _ { -4 } eV , finally increasing to 118 ^ { +9 } _ { -4 } eV for the final two observations . During one of these observations , we observe two phenomena which were previously suggested as indicators of quiescent accretion onto the NS : short-timescale ( < 10 ^ { 4 } sec ) variability ( at 32 ^ { +8 } _ { -6 } % rms ) , and a possible absorption feature near 0.5 keV . The possible absorption feature can potentially be explained as due to a time-variable response in the ACIS detector . Even so , such a feature has not been detected previously from a NS , and if confirmed and identified , can be exploited for simultaneous measurements of the photospheric redshift and NS radius .