A319 flight crew refused aircraft due to a malfunctioning cabin cooling system. Excessive heat caused passenger injury.

Narrative

Aircraft X has been written up on 4 occasions in the past 30 days for excessive heat being produced by the environmental control system. The arriving Captain explained that the FA (Flight Attendant) stated that the cabin temperature was excessive and needed relief as the temperature was increasing. The CA explained that he turned down the ECS (Environmental Control System) knobs to full cold but the FAs called back and said it was becoming unbearable and that the passengers were very uncomfortable. The CA switched off the 'Hot Air' switch which seemed to stop the temperature from increasing but the FAs called back and said they an elderly woman that was in distress due to the extreme heat. The crew [requested assistance] and the flight was met by paramedics. The elderly woman was removed from the aircraft by gurney. I reviewed the previous write ups and spoke with maintenance. They were getting ready to sign the aircraft off as good to go; unable to reproduce. I explained that a passenger had just suffered an injury due to a system malfunction and that this was not an acceptable solution to this ongoing problem. They said that maintenance management wanted it resolved and returned to service if they could not repeat the incident. It was signed off and I notified Dispatch that a passenger had just suffered an injury due to this system failure and that I was not willing to put an additional XXX people in an aircraft that had just cooked an elderly woman. They asked me if I was refusing this aircraft? I said that unless they fix this aircraft; I would not be flying it. They said they would look for another plane for my flight. I called union safety and told them of this dangerous situation and that it seemed that management refused to acknowledge the obvious fact that one of our aircraft was just involved in a passenger injury due to a system failure. He said he would make some calls. The aircraft departed for ZZZ 30 minutes later with a crew that was unaware of what had just happened. The aircraft experienced 2 more runaway hot air situation in subsequent flights and had a 5th and sixth write up for cabin heat uncontrollable.Cause: This is a dangerous situation that resulted in an injury to one of our passengers and yet this aircraft was repeatedly placed back into operations with only an 'unable to reproduce.' It is only by good fortune that more passengers and crew have not suffered injury by the blatant disregard for crew and passengers' safety. I tried to tell maintenance; Dispatch that this aircraft was not safe and nobody took a leadership management roll to protect our passengers or crew.