APU DOOR
From the B777 AMM which will be similar.
The air inlet door opens to let air into the engine and load
compressor plenum.
The air inlet door is on the right, upper surface of the tail cone
immediately aft of the rudder trailing edge. This position
decreases the possibility of water or slush ingestion during
ground and flight operations.
As has been mentioned most APU's now have a cool down mode prior to full shut down, this maybe approx 30 - 60 seconds dependant upon type.
Unusual to see the APU door open in flight but can be allowed fror despatch to sumplement the electrical system if a fault on an engine IDG for example.
The B787 may have extra electrical requirements that necessitate the APU being run longer than most, then being shutdown during the flight phase.
B787 course not on the cards yet.....