Basic question about respiration rate in mammals as temperature changes

Please correct my thinking if there is an error in it:

Body heat in mammals is created by “inefficiency” in the conversion of food to atp. The rate of cellular respiration in mammals must increase somewhat as temperature drops in order to provide additional heat to maintain body temperature. This is why more calories are consumed in colder temperatures.

A reptile’s rate of respiration would decrease as temperature drops, while a mammals would increase somewhat.

This is the impression that I have been under for a while, but I recently came across a source that stated otherwise and wondered if I hadnt been quite right in my thinking.