For what it’s worth; at one time we used to have frequent APU starting faults on A330 during a short taxi in at a hot location.
I don’t remember who came up with the fix, which was to push the master pb at about 8000’ while being vectored for the approach. The APU would then start normally and reliably first try on taxi in.
We supposed that venting the APU compartment helped in some way, but I like the fuel pump theory proposed earlier in the thread. Anyway, our fix worked very well.
PS, there is nothing wrong with mechanical sympathy. You wouldn’t select TOGA as soon as a main engine “Avail” light comes on or accelerate at full speed down the road as soon as your cold car engine starts, but I have seen many do the same thing to their APUs. I don’t have the photo to share, but once saw the result of people repeatedly restarting failed APUs without applying the recommended starter cycle cool down periods and also putting APU bleed and both packs on the second the “avail” light came on. The poor thing was a mess of blackened and knackered parts.
I start the APU, and when it is “available” I do something else for 30s or so. Then APU bleed on, do something else for 30s or so, then a pack on. Same in reverse when shutting down.