Sucks for the landlord's power bill, but the people who commented here can do better. Sorry to lump you in with them. I wouldn't want a landlord micromanaging my window use by shutting it themselves and demanding I keep it shut, especially if I had some valid reasons.

I suggested a dehumidifier because damage/mold was brought up, definitely wouldn't be very viable as you suggest (especially in regards to temperature). Thanks for keeping it grounded and fair, and you are right that hot air would be pulled in, but I'm not too sure about that having a major effect to the point of very expensive damage or major effects on power, especially if OP's door is shut and considering other factors, such as climate, room size, and it being a single window.

Some of the arguments here in the larger thread do make sense, pulling in hot air and another commenter suggesting that it would cause an effect across the house, pulling outside air from other cracks and openings and increasing humidity/temp throughout the building, but I feel that the effect on the house/system is minimal if it's a single window in a small room.

I have a weather app (Breezy Weather) that shows me a humidity graph, that could be helpful for OP if they are reading this.