We have found indications of a relationship between the differential emission measure (DEM) weighted temperature and the cross-field DEM width for coronal loops. The data come from the Hinode X-ray Telescope, the Hinode EUV Imaging Spectrometer, and the Solar Dynamics Observatory Atmospheric Imaging Assembly. These data show that cooler loops tend to have narrower DEM widths. If most loops observed by these instruments are composed of bundles of unresolved magnetic strands and are only observed in their cooling phase, as some studies have suggested, then this relationship implies that the DEM of a coronal loop narrows as it cools. This could imply that fewer strands are seen emitting in the later cooling phase, potentially resolving the long standing controversy of whether the cross-field temperatures of coronal loops are multithermal or isothermal.