I'm sorry, I should have read your original question more carefully, you did specify which one you meant. The flight crew rest air comes directly from the flight deck conditioned air distribution system. The only AMM for a 777 with that type of crew rest I have access to is for the freighter, and there the crew rest gets conditioned air from the supernumerary area distribution duct - so that'll be the minimum temperature then... For the passenger version I can't be sure, but I'd wager its flight deck conditioned air.