Check the original lease..I know mine states that the renter (me) is responsible for pest control. That means that when I signed the lease, I agreed to regular pest control maintenance.That was agreeable to me because the place was bug free when I took possession of it. Now if I came up with bedbugs or any bugs while I lived there it would be on me to eradicate them. If I simply allowed them to overrun the place and then moved leaving it bug infested for the owner to deal with then I obviously would be responsible for the cost of eradicating the problem but that wouldn't happen because I have maintained regular pest control as agreed. Check your lease. If you're not responsible for pest control then I don't see how he can consider that "damages" like he would a hole in the wall or ripped carpet, etc.