Possible and not uncommon reason for the open APU inlet door:
I have seen it happen that the APU ilet door actuator was inoperative. In order to be able to provide APU power if neccessary, the aircraft may opearate under an MEL, with the door secured in its open state throughout the whole operation and an extra percentage of the tripfuel accounted for the aerodynamic loss. So open APU inlet does not neccessarily mean a running APU.