B737 air carrier pilot reported excessive cabin temperatures when the APU was inoperative and improper ground equipment was provided for external start procedures. The pilot returned to the gate and deplaned the passengers.

Narrative

APU was MEL'd. We arrived at aircraft at XA:05 for XA:40 departure. Temp on aircraft was already 80-83 degrees and pre-conditioned (PC) air was not hooked up. I did the walk around and spoke the ramp about needing air start and asked to have the PC air be hooked up. I returned to the flight deck. Temp on aircraft stayed very warm through the boarding process and feel that they never turned on the PC air after hooking it up. Boarding was finished and the main cabin door was shut at approximately XA:40. PC air felt like it came on for a few minutes then off again as they disconnected preparing for engine start. The first of 3 engine start attempts was at approximately XA:50. While sufficient pressure was there at the beginning of start the engine never reached appropriate RPM. I called Maintenance and they confirmed via cameras that wrong airstart cart was being used. I notified operations via radio to tell the ramp they needed another cart. They said one was coming. At this point; XB:00; the temp on the aircraft was to 88-90 degrees and told operations to have them hook the PC air. Operations said sorry but it would just have to be disconnected again and wouldn't be worth it. I started getting upset at this point and told them to hook it up explaining the hot temperature. It was hooked up a few minutes later and it began to cool. While still waiting for the correct airstart cart at XB:15 the ground power quit as well at the PC air and the plane went on battery power; which means no lights other than emergency exit; no gaspers or recirculating fans for airflow so it got hot fast. I notified operations telling them they needed to get the jetway back to the aircraft to deplane the people. I informed them of the urgency of the situation and that they needed to consider it a [priority]. I was told they were working on it numerous times and that they were dealing with other aircraft and to just wait. I got more upset at this point. After numerous times trying to convey the urgency of the situation to ops but to no avail I called the chief pilot and notified them of the situation and that I was going to refuse the aircraft when we did finally get the people off. I notified Dispatch of the refusal as well. The jetway did not get to the aircraft and main cabin door open until XB:45. After deplaning was finished I notified scheduling and the chief pilot that I was unfit to continue on duty for the rest of the day.