Metabolic rate is measured in O2 sec^-1, it all depends on different animals. The surface to volume ratio, where the mouse have a huge surface compares to its volume ratio and an elephant have a small surface to its volume ratio will all affect on the heat loss
For the mouse, more heat are lost due to the large surface area and small volume, so they would have make up for the heat loss.
So the mouse affinity of oxygen will be less because the oxygen transferred from the lungs would have to be carried to the rest of the body and supply energy by allowing the nutrients to be burned (metabolism) whenever the oxygen are delivered from tissues.
The mouse overall will need more energy than an elephant because the amount of heat that are lost. According to Le chatelier's principle, whenever the system is temporarily out of equilibrium, then it will just simply shift back. In this case the mouse lost heat or energy, and eventual gain their energy back.
