There doesn't have to be anything in your lease that requires you to use common sense during your tenancy. Negligence is determined by a standard of care exhibited by a reasonably prudent person. When it's 15 degrees outside, a reasonably prudent person would keep the temperature up so that pipes wouldn't freeze. Think about that standard in anything you ever do about anything.
That being said, I've handled enough frozen pipe water damage claims to be able to tell you that leaving the garage door open for 30 minutes isn't likely to be the cause of the frozen pipes unless there is the remote possibility that a pipe in the wall of the garage froze..
It would be interesting to know exactly where the break in the pipe was. Most house water lines run through exterior walls and are exposed to the extremes of weather. If a pipe broke in an exterior wall, it likely has nothing to do with the garage door being open for 30 minutes.
So far I don't see anything to indicate that you are responsible for the damage and you are free to say no to your landlord. If he keeps your deposit at the end of your lease, you will have to sue him for it.
Frankly, I wouldn't offer any money in compromise and just wait and see what he does.
Meantime, do you have renters insurance? If not, now you know why you should have it.

