A320 Pilot reported the refusal of an aircraft due to the APU not working correctly resulting in increased cabin temperatures over multiple legs.

Narrative

Aircraft Refusal for APU Bleed and Inadequate jetway Air. This report concerns an aircraft refusal (Unable to Fly- UTO) for no APU bleed on Aircraft X and being unable to cool the cabin to acceptable levels for the PAX on the ground. The real issue is the inadequate jet way air all over our system. There was also an issue with ACARS not accepting my UTO write-up. We started off with this aircraft on the first flight of a two day trip and were scheduled to keep it for 4 legs. During preflight; we got an AIR APU BLEED LEAK ECAM. The APU bleed air was put on MEL and we proceeded to ZZZ1. The issue quickly became the hot weather at all our destinations (ZZZ1; ZZZ2; ZZZ3; etc.) and the inability of the jet way or portable air carts to keep the cabin cool enough. We struggled with this in ZZZ1 and back in ZZZ2; but managed to keep the cabin temp no higher than 80 before the PAX were boarded. Once the engine was started at the gate; we could immediately cool the cabin more effectively with the packs. However; ZZZ3 proved more problematic. The aircraft was supposed to be fixed overnight there and it didn't happen. We were also at a minimum layover of 9 hours behind the door due to WX in ZZZ2 the prior evening. When we got to the aircraft in ZZZ3 the next morning (gate X) for our 1½ hour late departure; it was 86 degrees and humid in both cabins with 80 degree air coming in from the jet way hose. The outside air temp was about 90. We notified ZZZ3 Ops that this was too hot to load our PAX (several were elderly; in wheel chairs; etc.; and the flight was almost full and we've all seen how the temps go up another 5-10 degrees once the plane is loaded without proper cooling). Ops agreed and they held off boarding and tried to adjust the jet way air. These are all city-owned jet ways (as with most places) and ops has little control other than turning them on and off. In fact; they mostly have no controls now for adjusting the temps and it's all supposed to be automatic; as long as the temp probe is in the cabin (it was). But evidently we can put men on the moon 50 years ago but can't solve the jet way temperature mystery for the 50 years since. It seems to be an ongoing engineering mountain and even with relatively new jet ways in ZZZ3; they just can't produce cold air for some unfathomable reason. After two different portable air carts were brought; we never got less than 80 degree duct temps or lowered the cabin temps from 86.I made the suggestion that we start the number one engine with the jet way attached; cool the cabin; and load the PAX with it running. We coordinated with ZZZ3 Ops; Dispatch; and the Operations and all agreed it was doable so long as everyone was briefed on the ramp and the PAX were spread out during the loading process so they wouldn't have to listen to the engine for too long and could just walk directly onboard. This worked and after 10-12 minutes; we notified ops they could begin boarding. As we finished our flight planning; I noticed they had kept us with this airplane for our upcoming ZZZ2-ZZZ1-ZZZ2 turn; which would make it 6 straight legs with this issue. That was a change as the original schedule had us changing planes once back in ZZZ2. Having already done the ZZZ1 turn the day before; I realized this plane needed to be fixed (and good thing as ZZZ1 was 42C when we got there later; or 108 degrees). After reviewing the aircraft refusal process in the FOM and an information sheet; I notified Dispatch en route to ZZZ2 that we were refusing the airplane; to give ZZZ2 a heads-up and time to find another aircraft as required. He acknowledged and said he'd start the process. I also asked if we needed to bring Maintenance or others in to discuss (per the guidance) and he said not required because he notified them already and they were in the process of coordinating its removal back in ZZZ2. I sent the code (I initially forgot the 'XXX'; but cancelled that entry and put in the right code). I also put in thecomments field the following: ZZZ2 ZZZ1 APU BLEED INOP..HIGH SFC TEMPS W/ INADEQUATE COOLING AIR FOR PAX." per protocols from FOM. But I got back an ACARS message saying that log number was 'not active for this aircraft'. I double checked the Maintenance Release (MRD) and our maintenance papers and it was indeed the right and still active log number and in fact the same one I put on the sticker next to the APU BLEED switch (that Maintenance read to me) the day prior. But the system said NOPE basically; and wouldn't let me send the UTO after a couple tries. I asked Dispatch about it and he said I could disregard the ACARS entry since 'it is entered in the system so MX/OPS has the info.' This seemed a little strange and I sent a follow-on message about the fact it said the log wasn't active when it was; and also the fact the UTO needs to be in the 'Unable to Operate' section at the top of the Log History Report for 14 days/20 items; to notify pilots on subsequent flights (also per Company protocols). I never received a response. At this point; we were approaching the top of descent and further communication efforts on my part ceased; to focus on the arrival. We landed in ZZZ2 without incident. However; while pre flighting in our new aircraft for the next leg to ZZZ1; our Rep in the Ramp Tower called to ask about the UTO. He said it hadn't shown I'd coordinated with MX for the refusal; which is part of the protocols; and suggested I could still contact them. I told him that was strange since I had asked the Dispatch when I refused the aircraft en route if we needed to bring them in and he said he'd already done the coordination and they were working on it; as mentioned above. He thought that should qualify as contact and not to worry about it. But clearly the ball was dropped somewhere."