I think there may be an error in the way heat is being calculated when a mission is completed and you return to Haven.
For example, in the mission "Rise or Fall," the heat that was added to my Skolver Cap during the Prize Wheel of the arena level was enough to make the cap half way through Heat Level 9. When I got back to Haven, my helmet was Complete (Level 10) instead of half way through Heat Level 9.
I was using a Heat Amplifier at the time, which could be relevant.
the heat from the amplifier is not shown at the end of the level so during the prize wheel the heat you see is what you would get if you didnt have the amp on