A321 Captain reported an APU auto shut down after departure. With one Engine IDG already on MEL; the flight crew elected to perform an air turn back and overweight landing rather than attempt ETOPS to destination airport.

Narrative

Aircraft X was delayed for maintenance. The APU was on MEL and on the inbound flight Gen 2 had failed. I talked with maintenance and the plan was to fix both. They were working on the APU and waiting on a part for Gen 2. Because Gen 2 would require an engine run; we held off boarding until this was complete. Maintenance finished their work and we started boarding. The APU was cleared but Gen 2 was still inop. We obtained Version 2 of the release with MEL 24-XX-XXX. Our altitude was lowered to FL330 per the MEL. MEL A22-XX-XX was also applied. I chose to run the APU during the entire preflight and boarding to make sure that it was good. I also added 1000 lbs. of gas to offset the APU burn on our 4+36 minute flight. We departed at XA38z or XB38 pacific. 38 minutes late. Taxi out and take-off were uneventful. We had a significant number of live TVs out perhaps it would be better to turn the TV system off when working on the aircraft electrical system. Passing FL320 our APU shutdown! We descended back to FL310 and attempted a relight. The APU restarted Hooray! And then auto shutdown about 10 seconds later. We were now single generator with no chance to get a second one online. I made the decision to return to ZZZ. We were only 30 minutes out and it had a long sea level runway with good weather. I notified the Flight Attendants with using the emergency call. I had started with all call but when only 2 answered I used emergency call. I wanted the entire crew on the call. The Flight Attendant 1 asked if it was urgent; and I said yes. We informed ATC that we needed to return to ZZZ. They asked if we wanted to request priority handling and I said yes do to the overweight landing. We were around 180000 lbs at that time we landed around 176000 lbs. I notified Dispatch via ACARS and he notified the ZZZ station. We completed the aircraft monitor and QRH. We also ran the overweight landing checklist. We landed at ZZZ at XC00 local XD00Z. We shot the ILS XXL. I intercepted the glide slope early and followed it down. I also configured early. While the overweight checklist mentioned turning the auto thrust off and slowing to get below the flap speed; this wasn't required at our weight. I used auto brakes low and max reverse. Max brake temperature was around 200 degrees C. Taxi in was uneventful and we parked at Gate five minutes after touchdown.