As was already said, all components of the atmosphere will have the same temperature. As the air cools or warms, this will remain true because the air is pretty well mixed.

However, you may be interested to know that temperature is not the same as heat. The "specific heat capacity" of a material or gas is the amount of heat required to raise its temperature by one degree. Earth's atmosphere is almost entirely made up of nitrogen, oxygen, and argon, with trace amounts of other gasses including carbon dioxide. A gas with a low specific heat capacity, like argon, doesn't need to lose/gain as much heat to change temperature when compared with a gas with a high specific heat capacity, like carbon dioxide.

Finally – say you add a particular amount of heat to the atmosphere. The proportion of that heat that goes into different gasses depends on both the specific heat capacity and the proportion of each gas in the atmosphere. A gas like carbon dioxide may have a high specific heat capacity, but because of its very low proportion in the atmosphere (<0.1%), it won't take a very significant proportion of the heat. In this way, I think the major atmospheric gasses all have about the same influence on changes in temperature (i.e., their proportions and specific heat capacities all very roughly balance out).