EMB-190 flight crew reported air conditioning system malfunction during climbout.

Narrative

This was the third leg on the aircraft with no APU. On previous flights the plane would get quite warm during the air start procedure and wouldn't really cool down until we were in the air (temps in Location and Location 1 were around 100 degrees that day). Once in the air the cabin temperature would drop quickly. There was a long taxi line and this time with the airplane so warm we talked to Operations about returning to a gate; however they were able to help us cut the line and back taxi for a faster takeoff. Once airborne the plane did not cool; but rather heat was pouring out of the vents. The cockpit temperature rapidly climbed to 43 Celsius (109 Fahrenheit) with no sign of the trend reversing; and cabin temps climbed to 35 Celsius. We asked to level at 7;000 ft. as it was clear we had no control over the temperature despite having both knobs set to full cold (and the ECS (Environmental Control System) page on the MFD (Multi-function Flight Display) showed we were set to 18 Celsius for all three zones). We decided to return to Location and declared for priority handling; as continued exposure to that level of heat was a threat to the safety of the crew and passengers. We also called Operations and advised them we would need an open gate with ground air immediately. The air return was uneventful; and we had some temperature relief once the descent was started; as higher thrust settings were forcing more hot air into the cabin. Once on the ground we taxied to the gate and deplaned. Tech Operations met the aircraft and the discrepancy was noted in the aircraft logbook.