Air carrier Captain reported repeat pack failure on climb. The Left Pack was already on MEL and the Right Pack temperature became uncontrollable. The flight crew descended and shut off the Right Pack and diverted to make a precautionary landing.

Narrative

Aircraft X landed earlier in the day at ZZZ with a maintenance issue. Talking with the Captain from that flight; he told me they had a L PACK TEMP EICAS Message with a PACK OFF and INOP overhead lights. They could not reset the pack using the QRH procedure; so [they] turned it off. He then said the cockpit started getting very hot. They couldn't control the temperature and it became so hot they eventually had to descend below 10;000 feet; depressurize and turn off the Right Pack.When we arrived at the airplane; maintenance was deferring the Left Pack. I asked about the temperature problem with the right that the previous crew had inbound. Maintenance informed us that they had tested the Right Pack and it checked normal. We started the APU and ran the Right Pack. Air was very cool and seemed to be working normally. After engine start and during taxi/take off/initial climb out; the pack still worked normally. When the flaps were retracted and the pack went to Hi-Flow; the air seemed to get warmer but not bad. By 10;000 feet the flight deck temperature was approaching 30 degrees. We continued the climb but eventually asked ZZZ Center if we could stop at FL260 while we worked an A/C temp control problem. By then the flight deck temp was up to 39 and slowly climbing. We tried turning the Flight Deck Auto trim switch to manual and holding it in the cold position. The trim air needle moved toward the cold position; but it just kept getting hotter.We turned the right pack control switch to N; but it got hotter. We turned off the trim air switch; still got hotter. By now we determined there was no way we could continue to ZZZ1; as flight deck temp was now 45. First Officer flew the airplane and took left comm (ATC communications) while I attempted to contact Flight Control/Maintenance on AIRINC. Phone patch finally went through but it was hard to communicate as the reception was bad. I was able to notify them of the problem and said we could not continue to ZZZ1 but needed to land somewhere else. We mutually decided ZZZ2 (by this time we were approaching ZZZ3). I felt ZZZ2 was a good choice. Flight Control ran the numbers for us and gave us weather and NOTAMs. I asked if we could get Name on the phone and informed him that there was a possibility of an overweight landing. Between Name; Maintenance and I; we determined the best plan was to go straight to ZZZ2 and land (overweight if necessary). The First Officer coordinated with ZZZ2 Center for a reroute to ZZZ2. We began our route and started to descend for 2 reasons:1. It was now 47 degrees C in the flight deck (117 F) and we needed to get below 10;000 feet so the Right Pack could be shut off.2. I wanted to get down and burn off more fuel to hopefully get below 43;000 pounds wing fuel and 283;000 pound landing weight.We went down to 8;000 feet; depressurized and shut off the right pack. We also extended the spoilers in an attempt to burn more fuel. By the time we passed landmark we were down at 4;000 feet; flaps 5; 200 knots with spoilers extended. This all worked out as fuel on board was 43;000 lbs and aircraft weight was 282500. We were handed off to ZZZ2 approach and asked for vectors to [Runway] XX final for the visual. We retracted the spoilers and completed the approach and landing normally.Of note; as we were descending; both Captain and First Officers iPads overheated and shut down. This was unexpected and not something I had been thinking about. I got out of the seat and put both in the catering cooler for about five minutes. Fortunately this worked and they both turned back on.