If two or three tenants in a row move out and cite sky high power costs due to poor insulation as the reason, the landlord will notice and do something about it. One tenant is a fluke, several all saying the same thing over a few years is a problem that will get dealt with. Nothing costs a landlord money like high turnover. But departing tenants DO need to communicate that as the issue, otherwise the landlord would never really know since s/he isn’t the one paying the bill.
In providence RI my friends and I rented a 3 BR 2nd floor of a duplex. The heating cost was insanity. The rent itself was 1350 per month and the oil heating in the winter cost us 3600, which is 100 per month added onto my 450 share for the entire year. I left that year, my roomies stayed and begged the landlord to switch, the oil heater was wicked old and inefficient. He said no, he didn't care. They moved due to that. So yes, it did cost him, probly the months worth of rent that he lost due to the realtors finders fee.