Final answer:
In the case of an amber APU SOV message after an APU fire shutdown, the flight crew should refer to the aircraft's Quick Reference Handbook and follow the APU fire shutdown procedure again, possibly including manually isolating the APU from fuel.
Step-by-step explanation:
When an amber APU SOV (Auxiliary Power Unit Shut-Off Valve) message appears after the APU has been shut down due to fire, and the aircraft's engine is supplying bleed air for air conditioning, the proper checklist action according to standard aviation safety procedures is to consult the specific aircraft's Quick Reference Handbook (QRH) or emergency procedure manual. Typically, the APU SOV is meant to automatically close to prevent fuel from feeding a fire. If the SOV message appears, it could indicate that the valve did not close as expected. In this scenario, the flight crew should perform the APU fire shutdown procedure again, ensure that the fire extinguishing system has been activated if necessary, and then follow any additional steps outlined in the QRH, which could involve isolating the APU from the fuel supply manually.