Generally, the nature of the damage is specified, rather than the cause. The answer by Jasper mentions "burst pipe", which is a common phrase used when a pipe freezes and is damaged, but a pipe can burst from reasons other than cold -- excessive pressure, which is more likely to come from heat than cold.
The boiler overheated, and the pipes burst as a result.
A board, or a sheet of metal, or a concrete slapslab, may crack from cold, but such things might also crack from an impact, say the blow of a hammer. A glass might shatter or simply break from the action of freezing water, but again those verbs can be used with other causes. "frostbite" is specific to cold, but it is a very specific type of injury. It can also be used for plants, and for injuries that can and do heal.
For an animal or human, one can say "froze to death" which is surely beyond repair. One might say that a plant has been frost-killed. Any living being may suffer hypothermia -- injury due to cold.