The issue was not about staying warm on the Sabbath. It was about kindling a fire on the Sabbath - gathering wood etc.
That is why God personally ordered a man killed for gathering sticks on the Sabbath day.
Exo 35:2 Six days shall work be done, but on the seventh day there shall be to you an holy day, a sabbath of rest to the LORD: whosoever doeth work therein shall be put to death.
Exo 35:3 Ye shall kindle no fire throughout your habitations upon the sabbath day.
Num 15:32 And while the children of Israel were in the wilderness, they found a man that gathered sticks upon the sabbath day.
Num 15:33 And they that found him gathering sticks brought him unto Moses and Aaron, and unto all the congregation.
Num 15:34 And they put him in ward, because it was not declared what should be done to him.
Num 15:35 And the LORD said unto Moses, The man shall be surely put to death: all the congregation shall stone him with stones without the camp.