B-737 Captain reported a safety concern where ramp personnel disconnected ground conditioned air from the aircraft with an inoperative APU resulting in excessive cabin temperatures. The Captain deplaned passengers and crew for passenger safety and refused to operate the aircraft.

Narrative

I was Captain operating Flight XXXX (ZZZ ZZZ1) gate XX. Initially the flight was planned to fly a 737-700; but a subsequent equipment substitute changed that to a 900 ER / delayed because our aircraft was arriving from ZZZ2 (on time). The delay was updated before boarding to on time because another equipment sub changed aircraft once more. This aircraft was scheduled to operate to ZZZ3; but was just refused by the Captain because the APU was inop and the aircraft was extremely hot.In the gate area; ZZZ3 passengers were deplaning and I was made aware of the APU deferral. I told the Flight Attendants that I would walk through the aircraft and assess the temperature. It was hot; but I had also walked outside and discovered the gate PCA air hose was kinked 90 degrees in 2 places and the hose was improperly connected to the aircraft. I personally fixed the hose. I returned to the gate area and called ZZZ1 station operations to ensure they had an operational air cart for the next crew - they said yes.Next I took the flight attendants down to the aircraft and explained that I fixed the air hose and wanted their opinion about the temp as we opened the air vents and closed all the window shades. They agreed that it was uncomfortable but not dangerous. We all agreed to operate this aircraft to ZZZ1. I returned to the gate area and informed the passengers via gate PA about the first Captain refusal; the hose fix; the evaluation of our crew regarding the cabin temp and I determined we will proceed with this aircraft to ZZZ1. NOTE: many of these passengers were cancelled the night before and slept on the terminal floor waiting to get to ZZZ1. During the preflight and boarding process approximately one hour before pushing I chat to station operations; then called them on the radio. Discussed in depth the requirement for an air cart to be near / ready and to keep the gate PCA air cold and connected as long as possible prior to air start. Operations Control replied that the air cart was at gate XY and will be ready.This was not communicated to the ground crew. In fact a ramp worker walked over and shut down PCA air; counter to my communications with Operations Control. This began the cabin heating up rapidly; I opened the flight deck window to instruct him to restart PCA air ASAP. This while he was giving me electrical disconnect hand signal. Complete and utter communication failure. I repeatedly press the ground call horn to ensure this new tug driver knew this was a no APU air start; and asked the status of the air cart. It took 15 minutes to position the air cart. This air cart was not operational or the crew did not know how to operate it. More coordination and radio calls to Operations Control asking to find a new cart. This took an additional 10 minutes. This second air cart started but the ground crew did not know how to operate the cart; we heard the cart spool up but had no duct pressure to start the engine. The temperature in the aircraft became unbearable; then our ground power was disconnected and the jet went black.At this point it was clear to me that this rapidly degrading situation became a safety issue for the passengers with heat in the cabin. I called Operations Control and instructed them to return the jet bridge to the aircraft immediately- I told the flight attendants to disarm doors and deplane the passengers immediately for their safety. I proceeded directly to Chief Pilot's office debriefed the chain of events and multiple levels of failure and total lack of communication with the ramp crew. I returned to the gate area to ensure all passengers had been deplaned; and evaluated the temperature in the cabin - which was well beyond the safely limits; therefore I refused the aircraft via dispatch and the Operations Manager.