Air carrier B767 First Officer reported the aircraft air conditioning system failed to cool the flight deck and avionics even after maintenance efforts; resulted in aircraft refusal.

Narrative

On Day 0 I reported at XA35 for a XC05 departure ZZZ-ZZZZ. After briefing; we arrived on the jet a few minutes before XB00. As we entered the flight deck; we all noted how incredibly hot it was. We started the APU to hopefully cool everything down. I unpacked and built my relief pilot nest before leaving the flight deck for my walkaround. Usually upon return from my walkaround; the flight deck is noticeably cooler. This night however; it was noticeably warmer. Our equipment cooling system auto function was on MEL and the standby mode was not keeping up with the heat created by the avionics. The flight deck ambient temperature was over 100 and the heat coming off the panels was over 120 (one of us had a thermometer.) None of us could sit in the flight deck. Unfortunately; we were boarding; there was no where for the flight deck crew to sit and wait. We alternated between trying to stay out of the FA's (Flight Attendant) way in the forward galley and trying not to look awkward in the jetway whilst passengers were boarding. Once passengers were fully boarded; the flight deck was still over 100 and Maintenance said we just had to wait it out" for the flight deck to cool down. Once again we're left standing in the jetway in suboptimal air conditioning. This is around XB50. Waiting; standing and waiting; standing and waiting. I noted here that I was heat soaked; tired; and making quite an effort to not be snippy with Maintenance or Ops (this should have been my first clue that I was sliding toward fatigue; I'm normally a VERY bubbly and happy person.) Every so often one of us goes to the flight deck to see if the temperature has changed at all and chat with Maintenance. Barely changing even though they have a pc air hose blowing straight into the equipment bay. It was warm enough that even Maintenance wouldn't sit in the flight deck for more than a few minutes at a time. Finally after much discussion; Maintenance and ops decides that we need to deplane. This is now around XD35. Once all the passengers are off; we have some more conversation with Maintenance and finally settle into a few first class seats to wait around XD55. Since we had had all the passengers on the plane; the cabin was still quite warm. I sat down and immediately noticed an overwhelming feeling of lethargy. I felt like a wilting flower. My eyes were stinging and I was trying not to rub them lest I smear mascara all over my face. I tried to put my head back and catch a few minutes of rest; but the warmth in the first class cabin mixed with the FAs walking up and down the aisle prevented any real rest. Around XE35; the CA (Captain) asked me a question and I had to say "huh?" multiple times and ask him to explain what he was asking. He asked me how I was feeling and I started cataloging my symptoms. After some slow thought; I recognized that my personal symptoms of fatigue had already set in: lethargy; headache behind my eyes; stinging eyes; trouble processing verbal communication; and irritability. I made the decision to call in fatigued. When I packed up my nest; I almost forgot my iPad (that was sitting right next to my flight kit) and my headset (also in plain view.) By the time I made it to the field hotel; I was tripping over my own feet walking. After sleeping at the field hotel; I went to brush my teeth the next morning and realized I was so tired that I forgot to brush them the night before. When my husband called that morning; he noted that I still sounded exhausted. I ended up sleeping 10 hours last night (Day 0 into Day 1). I'm still a bit tired 36 hours later as I write this.This was all compounded by the trip I had before this one where we left ZZZ for ZZZZ1 late (due to Maintenance and a last minute a/c swap) and once I got to the ZZZZ1 hotel; my a/c unit was stuck in heat. The front desk sent the engineer up and he changed it to a/c. The room cooled quickly thereafter; however this ruined my opportunity to take a long nap and instead I only got about 90 mins. Overnight; the a/c kept resetting to heat and it woke me up twice with the room over 76 degrees. Once back in a/c; it would cool very quickly. This; plus the all-night flight over to ZZZZ1 started the fatigue ball rolling down the hill. I slept ok on the night in-between the trips. 6 days out for me is a non-issue in comparison to a former life as a corporate pilot who was on the road for 15-16 days in a row."