radiant fracture

Yes, I'd agree with this. Further, your landlord presumably already knows how much it costs to heat your Shed, given that he was paying the bills himself for years, so he's unlikely to be surprised by a request to help reduce them. Asking for one modest but specific improvement, like sealing up the gaps around the doors, seems perfectly reasonable. You can also present it in terms of wanting to cut carbon emissions and protect the environment if you're embarassed about seeming too much motivated by your own pocket. I would be having a proper look inside that cupboard, too, where you may well find there's an obvious gap you can plug yourself.