Captain reported the APU shut down while at a holding point away from the gate causing the aircraft to be left without power. Despite attempts to alert the company for assistance; the crew and passengers were left in the heat with no power; lights; nor air conditioning for 40 minutes.

Narrative

I was the Captain on this flight. Scheduled departure time was XA:28. After most of the passengers were boarded; we received notification that flights to ZZZ1 were going under a delay program due to weather in the Northeast. We were given an estimated departure clearance time (EDCT) of XD:38 (over 3:15 later). After discussions with Dispatch; Operations; and the gate agent; we made the decision to de-plane the passengers and re-board them at approximately XC:25. Operations informed us that the gate had to be vacated by XD:00. We pushed off the gate at XC:51 and taxied to a holding point on Taxiway 1; short of XX for Runway XXR. Since we were over 30 minutes until the EDCT; we informed Tower that we would shut down both engines to save fuel and would re-start to be ready to meet the EDCT time. Just prior to starting the #1 engine; the APU initiated an Emergency Shutdown. After allowing for a recovery/stabilization period; we attempted to re-start the APU and another shutdown occurred. A third and final attempt to start the APU was attempted with the same result. At this time; with the batteries as the only power source; we informed Tower of our situation and requested assistance. Operations was contacted and informed that we needed a ground power unit; an engine start cart and a ground air conditioning unit. The response received from Operations was; 'Ramp is really busy right now and it may take some time.' I then tried to convey the seriousness of the situation because I felt Operations had a lack of understanding of how quickly this situation was about to deteriorate - and it did! Shortly after talking with operations; we lost all electrical power to the aircraft. The only means of outside communications was via cell phones. I tried to again contact Operations via the phone number listed. I made two attempts at contact; and both times the phone calls went unanswered. I then called Dispatch and informed them of the situation and requested help in contacting any help at ZZZ. Luckily; we had an Check Airman as a jump seater who had contact numbers for ZZZ Tower and he assisted in contacting Tower for help. The duty pilot was contacted as well. I contacted Maintenance Control and asked for any guidance they could provide towards possibly recovering the APU. There was a procedure attempted; but it did not recover the APU. During this period of time; the situation in the cabin became worse. Internal temperature began to quickly rise; and with the 'dark' conditions; babies crying; and hot; irritated; and a few panicking passengers; the Flight Attendants had their hands full! Since we saw no evidence of assistance arriving; we decided to de-arm and open 4 passenger doors; and station a Flight Attendant at each open door so that no passengers or children could access an open door. Again; using my cell phone; we requested air stairs; buses; and as a precaution; an ambulance from ZZZ Ramp Control. The controller was very receptive to our situation and tried to help coordinate assistance with our request. After approximately 40 minutes; Ground and Maintenance personnel arrived with the requested equipment. ZZZ airport rescue and firefighting also arrived in response. The decision was made to start the #1 engine and taxi the aircraft back to the terminal under power. The air stairs and buses never showed up. The aircraft was taxied back to the gate without further incidence. Passengers were de-planed without any injuries or further problems. Because of all the events that took place; the Flight Attendants did not want to continue with the flight. An attempt was made to change aircraft and secure replacement Flight Attendants; but due to crew duty day complications; the flight was cancelled.I have tried to convey the situation as best as possible; but I don't believe the preceding description does justice to how serious a situation we were dealing with; and the lack of understanding and assistance we encountered from numerous agencies. We had a 'dead' aircraft in hot conditions and nobody seemed to care! It is hard to believe there is no provisions to provide a 'rapid response' to situations such as ours. We were literally scrambling as a crew to ensure the safety of the passengers and trying to convey the seriousness of our condition. There were some very bright spots during this event. The First Officer performed beautifully; supporting my decisions and providing essential assistance. The Check Airman was a huge asset with his knowledge and experience. The Flight Attendants were exceptional! They maintained control of a rapidly developing dangerous situation. The Ramp Controller was also great in that she seemed to be the only person outside of the aircraft who grasped the seriousness of our situation. In conclusion; I was frustrated and disappointed with the response and apparent lack of understanding from numerous agencies.There needs to be a sense of urgency demonstrated by support agencies for these types of situations that was not demonstrated for us.