I"m afraide to mistake, but yesterday I try to check on the EICAS APU Maintenance Page (CMC) how long it takes the door to show OPEN (13deg) when going from OFF to ON? Nothing indication, absolutely! No N1, Door open, EGT...
My mistake. I forgot to mention that (normally) the APUC is not powered until you momentarily go to START. The APUC provides information to EICAS.
However, you might be able to power up the APUC with the Ground Maintenance Test Switch on the overhead panel prior to or during APU start (sorry, my memory is a little vague on this). Unfortunately, whatever happens, the APU has to run though a self test before it can start the APU. This may lead to delays in the parameters appearing. By that time, the door may already have already gone well past 13 degrees. According to my notes, this is the trigger for the OPEN indication on the maintenance page.
Unfortunately, both McHale's and my notes have some serious contradictions in them. It would be a step forward, however, if someone can prove the APU can start without the EIU's being powered before moving the APU selector away from OFF. The only possible signal available to the APUC to initiate the starter motor would then be from the 13 degree switch.
Note that the door opening relays are controlled by both the ON switch and the APUC. This is why you can open the door prior to powering up the APU Controller.