Back in 2005, I experiment w/ cutting down the heat in the house (due to serious $$ debt), back then, I wasn't a member here and I don't know much about bunnies. The person I spoke to say even a -20 deg. C, these buns don't mind.
So the house is at about 10 deg. C or so. Then on Dec. 27, 2004, it's a chinook week so I let my bun Little Happy to hop outside. Turns out his penis touch the soil and has a bacteria infection. But I didn't know until a few days later.
The vet treated him w/ ABX, and he's been living w/ me in an oil heater room since after. But to this date, I have my doubt that his own body's immune sys. may have been able to fight off the bacteria if the dining room (the place he likes to stay) is warm. A few mth. later, the bacteria is cured, but 3 mth. later, he died of liver cancer, so I always blame myself for the above incident.
What you want to do is up to you.