Actually, there is a good reason the hot water line broke first. When water is heated, air is pulled out of the water which makes it less spongy when it freezes. As a result, hot water pipes freeze, expand and break at a faster rate than cold water pipes.
Since hot water is not on all the time, the water in the pipe never stays warm for very long after it is run, making it just as susceptible to freezing as a cold water pipe.
To prevent frozen pipes, there are a couple of things you should do. First, insulate the pipe. You can do this with foam tubes, fiberglass tubes and/or fiberglass pipe wrap. If you notice that the same pipe freezes all the time, then you might want to consider having that pipe re-run through a warmer section of the house. Also check for areas outside where cold air might be getting in to the area where the pipe is freezing, as some additional weather stripping or insulation may be needed in those areas as well.
If your pipes are frozen, electric heat tape can be used to help thaw frozen pipes. I must caution you though; heat tape can be dangerous if improperly used. Be certain the product you have has a built-in thermostat, is UL listed and properly installed. A common mistake is to wrap the tape on top of itself as it goes through the pipe. This can cause over heating and potentially a fire. Read and heed all manufacturer instructions when putting the tape on.
If the pipe is still frozen, you may need to hire a plumber with a pipe thawing machine, or rent one yourself from a local rental company to thaw frozen pipes.