We have monitored the transient neutron star low-mass X-ray binary 1RXS J180408.9-342058 in quiescence after its ∼4.5 month outburst in 2015. The source has been observed using Swift and XMM-Newton. Its X-ray spectra were dominated by a thermal component. The thermal evolution showed a gradual X-ray luminosity decay from ∼18 × 1032 to ∼4 × 1032 (D/5.8 kpc)2 erg s-1 between ∼8 and ∼379 d in quiescence, and the inferred neutron star surface temperature (for an observer at infinity; using a neutron star atmosphere model) decreased from ∼100 to ∼71 eV. This can be interpreted as cooling of an accretion-heated neutron star crust. Modelling the observed temperature curve (using nscool) indicated that the source required ∼1.9 MeV per accreted nucleon of shallow heating in addition to the standard deep crustal heating to explain its thermal evolution. Alternatively, the decay could also be modelled without the presence of deep crustal heating, only having a shallow heat source (again ∼1.9 MeV per accreted nucleon was required). However, the XMM-Newton data statistically required an additional power-law component. This component contributed ∼30 per cent of the total unabsorbed flux in 0.5-10 keV energy range. The physical origin of this component is unknown. One possibility is that it arises from low-level accretion. The presence of this component in the spectrum complicates our cooling crust interpretation because it might indicate that the smooth luminosity and temperature decay curves we observed may not be due to crust cooling but due to some other process.