I would tell him I was fine with it, but that he is responsible for paying the extra heating bill as well as replacing anything which may be damaged or stolen by leaving the door open.
If he agrees to it, then great - turn up the heat.
Shitty heating only works comfortably at maximum with the door closed, dropped ten degrees when he started with this crap. You can always tell someone that they're responsible, but when it happens how are you meant to enforce that kind of stuff with anything other than intimidation? (not where my skills lay)
If he agrees to it, then great - turn up the heat.