@HercwasanEnemyofEducation
If that happens when camping you wear more clothes/foil blankets etc the next night to ensure that you stay warm.
WE DON'T KNOW WHY SHE IS SO COLD. SHE MAY BE WEARING NOTHING AND USING A THIN DUVET. Other options should be explored first before heating. Dd needs to be punished if she's ignoring parents and turning it on.
Sorry but more clothes/foil blanket doesn't work (seasoned camper). The best thing is (expensive) 4 season sleeping bags, and even then, I cannot tolerate very low temperatures. As a result, I won't camp below a comfortable temperature, because no many how blankets/clothes etc I cannot sleep if it is too cold. They are not sufficient for a coldness that is so severe that it wakes you from your sleep.
In any case is surely up to the parent of this child to ensure that the duvet that they provide is warm enough, and the pyjamas they provide are warm enough and worn. Somehow I don't imagine the child is sleeping naked without a duvet and wandering around the house at night putting the heating on. It's a very big assumption that this would be the case.
The difference between so cold that it wakes you in the night and you cannot get back to sleep and warm enough is not a blanket or a pair of socks.
Why on earth is the parent not taking responsibility for providing her with an electric blanket or a portable heater if they don't want the heating on.
OP has not offered any reasonable solution. She focuses on the heating bill, but seems to have no care for the wellbeing of her child.
Again, I cannot imagine even considering punishing my child for being cold. I would personally be horrified that I'd allowed them to be cold, and would be apologising and finding an acceptable solution as a matter of urgency (along with a checkup at the GPs if I felt it necessary).