Hi Nickos,
Its just the detection system that is U/S, not the APU on fire. But in saying that, why is it U/S and do you want to run the risk?
Follow the QRH, and shut it down as soon as possible is my opinion. (787 pilot, not flown the 777)
Do you have enough electrical power from 1 engine?
do the flaps/slats run at normal speed?
do the flight controls still operate in Normal mode?
If so, shut it down.
if not, you may need to keep it running until landing then shut it down on the roll out.
just my opinion