The auto HVAC module is connected to the info displays via the CAN network. Modules on the network self-monitor and will take themselves off the CAN network if they determine too many transmitted errors to prevent taking the network down. This resets after an amount of time. This of course assumes the mode of failure in the module allows its error prevention logic to still function as intended, so network errors can get tricky.
The air mix actuators (temp control) are controlled directly the module itself, but since the AC clutch is controlled via communication over the CAN network, I could see a CAN failure set in the HVAC module to cause a default to a set temperature. (Or maybe it will blow hot, but not cold since no compressor clutch control)
To me, the most likely cause reads as some sort of internal failure within the HVAC module. However, as stated previously, network errors can sometimes get wonky with a faulty downstream module corrupting the bus, faults with in the wiring, etc. Throwing a module at something based on limited info is peak parts cannon, and proper diagnosis to eliminate an edge case would require a very motivated DIYer.
I would start by reading the codes stored in all the modules, and go from there. If you don't have a scan tool that can do so, FORSCAN is free, and the OBD adapter required is not too expensive. Useful to get for these cars regardless.
Maybe someone else who has experience with this issue will be able to chime in.
