B2N2 I'd be asking the same questions as you. What parts of the APU assembly exactly, and by what temperature difference, get un-soaked in those 6-10 minutes? My first impression is it is one of those "look busy, Jesus is coming" ideas.
I did receive a grease guru advice in this direction, however. On one particular APU that had a history of 30% failed starts, after ECB change did not help. The suggestion was to open the inlet flap during descent and begin ventilating the whole compartment. Then attempt a start just before landing while still airborne, hoping for some ram effect help. (GND OAT 35-ish).
Though we did manage a string of successful lightups this way, the general agreement among the tech crews was, for all we knew, it might have been the goat sacrifice and curses that actually did the trick.
After two more component changes that did not help, that particular unit went u/s about 10 days later and ended up being replaced altogether.