B737 First Officer reported an air conditioning pack failure caused extreme heat in the cockpit so they diverted to a nearby airport.

Narrative

We had a completely normal pushback and engine startup. However; after engine start I felt the temperature in the cockpit rise. I mentioned to Captain that I was cooling off the cockpit by selecting a lower temperature. We taxied out to the runway and both made a comment that the air temperature in the cockpit was still warm but figured that the air temperature would cool off or stabilize once we got in the air. It is not uncommon or unusual to find that the cabin temperature selectors can be sensitive in the 737 and figured it would cool off shortly. After takeoff Captain again mentioned the heat and I tried yet again to adjust the cabin temperature. We turned the Autopilot on and tried maneuvering the selector in different directions in case the pack was frozen. At this point the cockpit temperature was showing around 50-60°C. We decided to wait a few minutes just in case the pack just needed time to adjust; but instead the temperature kept increasing as we climbed. As there was no cockpit indication or light showing that anything was wrong with the pack; we checked the unannunciated section of the FM (flight manual) non-normals but were unable to find anything regarding the overheat of a pack. Together; we discussed our options as the temperature kept increasing to a maximum temperature that I saw to be about 78°C. I began to worry that the electronics/avionics would start melting or that one of us would eventually become sick or pass out from the heat. Our discussions escalated very quickly from this will be an uncomfortable flight" to "perhaps we think about ZZZ [airport]" to "let's get this plane on the ground as soon as possible". We were completely drenched in sweat and I personally found it increasingly uncomfortable to breathe in the heat. We closed our air vents to try to prevent more hot air from blowing. We reached a top altitude of about 23;000 ft. and we told ATC we would need to be diverting to ZZZ. Captain handed me the controls and radios while he contacted the flight attendants; dispatch and notified the passengers. We followed the appropriate overweight landing guide in the FM and set up to land on [Runway] XC in ZZZ. Due to the overweight landing and possible brake temperatures; we asked ATC to have the fire trucks stand by just in case. Captain had an excellent smooth landing and we were careful to use all of the available runway for the brake temperatures. On touchdown the Yellow caution maintenance light appeared. We got off the runway and had the firemen measure our brake temperatures which they read back as normal and we proceeded to the gate. We started the APU and turned off the engine bleed air and left pack. Upon arrival at the gate we opened our windows and after finishing our checklist we opened the cockpit door for airflow. As passengers were deplaning they too were making comments about the heat as it began to heat up the forward galley. After the deplaning process we spoke to maintenance and wrote up the overweight landing and pack runaway. We debriefed the flight together."