Again, many thanks to all who replied.
Having a bit of time on my hands this morning, I had a dig around and found that someone raised this question on another site back in 2010, having seen a 787 demo flight at Farnborough. And some of the responses were downright fanciful!
However, Aviation Week had an article on the APU in 2013, reporting that the operation of the APU has its own problems. It seems that if the door is closed after shut down, heat continues to build up in the APU, causing the rotor shaft to bow after 20 minutes. After 2 hours, the shaft will straighten itself out. Boeing issued a note advising the APU must be selected 'On' after landing, in order to keep the flap open to allow cooling. Unfortunately, this can be done only with ground power connected as leaving the APU 'On' will drain the lithium-ion batteries within 15 minutes.
The full article is here:
Boeing Tackles 787 APU Heating Issue
Incidentally, watching 6 of these leaving Heathrow yesterday, 4 had the flap open and 2 (both BA) had it closed.
Thanks again to all.