A321 pilot reported on climb received an ECAM ELEC Gen 2 fault; tried resetting but failed to reset. They started the APU it ran for one minute then it auto shut down. Flight crew diverted and landed uneventfully.

Narrative

On initial climb out; aprox FL 240; received ECAM ELEC Gen 2 Fault; leveled off at FL 250. Complied with ECAM and attempted to reset the generator; it did not reset. Started the APU; initially came online; thus providing a second electrical source. However; after approximately one minute; APU Auto shutdown with no faults. We attempted a second start with the exact same result. Therefore; with only one electrical system; we coordinated with dispatch and diverted to ZZZ. Landed just under max landing weight; FO landing was smooth and uneventful. Taxied to a gate in an attempt to fix the aircraft and press on to ZZZ1. Coordinated with local contract maintenance and maintenance control. We assisted the process by running the number 2 engine in attempt to get the engine generator back online. The engine generator did not come online so maintenance control was going to MEL the engine driven generator. The APU kicked out a low oil pressure code as the cause of both inflight auto shutdowns. Oil quantity was checked and I was told the oil levels were normal. At that point; maintenance control was going to dispatch the aircraft with the engine driven generator on MEL and use the APU inflight. Since the oil quantity was normal; and the APU just had two auto shutdowns in flight for low oil pressure; I was concerned over the reliability of the APU inflight. No other troubleshooting was completed at that time. Therefore; I refused the aircraft. Also; using the new logbook procedures; I did NOT put a 'Capt refuses the aircraft' entry into the logbook. The overall divert went well; in my opinion. We had enormous amount of support from dispatch; ATC; maintenance control; ZZZ contract maintenance; and the gate agents that assisted the passengers. However; the recommendation I have; and something I will do from now on if refusing the aircraft; make a logbook entry should so that other crew members can get additional input that may not normally get passed on. In my situation; as I stated earlier; I was told the APU oil quantity was in normal range so it didn't make sense why there was an oil pressure problem with no other trouble shooting; so I refused the aircraft. The following day; I saw the logbook entry showed oil quantity was low and therefore serviced. With an info item refusal entry I may have provided the oncoming crew a little more information they could have used in their decision to sign for the aircraft.