In 2008 when the housing market crashed I got stuck way, way underwater on a mortgage. To avoid foreclosure I rented out the house (with a professional property manager), and I was shocked at the senseless destruction and idiocy of people. 3 times in a week the tenant complained that the furnace wasn't working. Each time a guy came out and said "the temperature on the thermostat is the temperature in the house. everything is working perfectly." and then billed me $120. Yet, each time the tenant complained (because he didn't believe the guy I guess??) the property manager had to send someone out. In retaliation for me not "fixing" the furnace (that was working perfectly) the tenant beat the living hell out of the walls and wood flooring, broke glass, and even put the knobs from the stove into the garbage disposal. Never again. I'll just bull doze the next house before renting it.
For future reference, that was a "inhospitable" scam (Aka uninhabitable scam.)
In many states (at least the ones that get cold) the lack of a working furnace means that the unit is not "hospitable" and the tenant is allowed to withhold rent until the unit is made hospitable. In some states, the tenant is not required to pay rent for the period of time the unit is not "hospitable." Your tenant tried to do that, but failed.