Re: This should be considered safety critical software
Well theoretically yes, but it isn't like your house can suddenly get cold enough to kill you while you are sleeping. You'll know the thermostat has gone out.
I'd be much more concerned with pipes freezing, which can happen while you are sleeping if it is very cold outside. Kitchens in particular often have pipes against exterior walls, and wouldn't take much of a gap in insulation for a pipe to freeze when it is below zero outside even though your house might still be in the low 50s when you wake up (i.e. wake up not dead, just upset when you get the heat working again and your kitchen floods)