And what did the captain do to stop it?? Or did it stop on it's own too?
Sounds like a checklist failure to me. Before I put power on an a/c (especially the APU) I check all the switches are in the correct positions.
To start the bus at least 2 switches need to be selected (more if you include bleed air, etc)
If he was adamant I would have pulled the FDR (or QAR) for analysis. If the EGT was still rising after a couple of minutes that sounds like you had a tailpipe fire (probably minor as it eventually went down) so why didn't you check and blow it out? Apart from the usual checks to confirm correct operation, I also hope you checked the FADEC for any excedances before releasing the a/c.