Although I am not an aerospace engineer, I feel I could at least fuel some discussion. The answer would be : it depends.
If A/C & heating is still operating and can cope with some increased flow (after all the cabin pressure is regulated through a bleed valve), temperature control would still be possible.
Otherwise, let us assume an adiabatic expansion of the air in the cabin (probably some sort of worst case). Starting with a cabin altitude of 8000 ft (2438m), the ISA pressure would be 75264 Pa, temperature=20 deg C (293K). Depressurizing to 41000 ft (12496m) would drop the pressure to 17874 Pa. The adiabatic expansion yields T.P^(1-1/gamma)=constant, hence the final temperature will be = 293*(17874/75264)^(1-1/1.4)=194K=-78deg C.
This is well below the ISA temperature (-55degC) at this very altitude. Maybe the temperature would not drop below -55 degC if there is still some venting available, but the SLF would still be a little stiff in the end.
Hope this helps