What should be verified if the APU fails to start after checking the fuel supply?

Verifying the electrical connections is crucial when the Auxiliary Power Unit (APU) fails to start after ensuring the fuel supply is adequate. The APU relies on electrical power to initiate the starting sequence, which includes activating the starter motor and engaging various control systems. If any electrical connections are loose, damaged, or corroded, they can hinder the flow of electricity, preventing the APU from starting properly.

The other options, while they may be relevant in different contexts, do not directly address the immediate need to diagnose startup issues related to the APU. Oil levels, for example, are important for the proper operation of the APU but incorrect oil levels would typically not prevent the APU from starting; it would more likely affect performance if running. Air conditioning settings and propulsion systems are also not directly involved in the initial starting process of the APU. Therefore, ensuring that all electrical connections are secure and functional is the most critical step in troubleshooting the APU start failure.