Climate scientists are concerned with deviations in the average global temperature, but what does an average temperature mean from a thermodynamic perspective? Temperature is inherently a local measurement of molecular kinetic energy, and therefore an average temperature across a large region seems to have little meaning in a thermodynamic context. At best we could try to demonstrate that global average temperature is a predictive metric for some other variable of interest, but the burden of proof would be on us.
I could calculate a weighted average by volume temperature of my car by measuring the dimensions of various devices and parts in the vehicle along with their temperature and then taking their sum weighted by their proportion of the car's total volume. For example perhaps the air in the cabin is measured to be 25 °C, the air underneath the hood measured to be 50 °C, the temperature of the oil is 200 °C, etc. After taking all of the appropriate temperature and volume measurements I might calculate that the average temperature of the car is 100 °C. But what does that tell us? Is that a useful metric?
Now assume that I told you that the average temperature of the car increased to 101 °C. Should you be concerned? Well your first question might be where did the temperature increase, and when/why did it increase. If I told you the air in the cabin increased in temperature to 40 °C you would be correct to be alarmed, since that temperature would be uncomfortable or perhaps outright dangerous to the occupants. However if I told you that the temperature of the oil increased to 250 °C you may then ask when did the temperature of the oil increase? If I told you it increased when you put your foot down of the throttle for a period of time then you would not be alarmed at all since it makes complete sense for the temperature of the oil to increase when the engine is throttled. We can see here that the average temperature of the car contains little useful information.
From this metaphor we can see that an increase in average temperature weighted by area/volume is not necessarily useful, and that we need more information to understand what is really going on. Has there been any research in this area?