I hope someone can double-check my conclusions. Ah, but look more closely at what you are citing, and notice that specific heat capacity is shown changing as a function of pressure, not temperature? Also, notice the pressure scale is shown in decade/logarithmic, to better show any changes at all by compressing the chart horizontally? Without that compression, the chart would be very very flat near the 1 bar or atmosphere mark.
And here are some reasons why I think heat capacity of air is not important. 1) the changes in pressure across this Sandia cooler system are very small, almost negligible, especially when you see how it takes a whole lot of pressure change to make a very small change in heat capacity. 2) temperature is not a factor that affects heat capacity of air. 3) Even changing the heat capacity of air for this cooler shouldn't really affect how it works, the neat parts of it.
The heat capacity depends on pressure. We'll discuss the pressure for the fluid bearing part, and the pressure for the spinning fan fins part, and tie it back to heat capacity. Regardless, just looking at the chart for heat capacity, you see that even some changes in pressure would still lead to almost no change in heat capacity.
For the fluid bearing, I think it's the fluid-dynamic type where pressure would be rather constant, therefore making the heat capacity rather constant. But even if the fluid-bearing is hydrostatic, because the fluid is so turbulently mixed around in there, it can easily transfer heat between the surfaces, so we wouldn't even care if the heat capacity varies inside. And even if the pressure varies, we already see from your chart above that the varying pressure still doesn't really change the heat capacity. So for the fluid bearing, it doesn't really matter.
For the spinning fans, we see that the pressure would somewhat increase at the fan blades, increasing heat capacity across the fan. But that's a good thing, because the air could then absorb more heat, though I still think it would be a small increase anyway.
More specifically, I believe the very cool effect is how the sandia cooler changes the flow behavior, the arrangement of the laminar/turbulent layers etc. (someone else can use the proper description) where the air meets the surface of the blade. So it's not that the air changes its specific heat capacity. Rather, you get a very nice effect even with the same specific heat capacity, because the way the air interacts with surfaces is changed due to how layers are formed under the pressure of spinning/centripetal forces.
Notice that neat effect is lost in traditional coolers, because the spinning fans in traditional coolers are just plastic blades that aren't cooling the air, just moving it. The traditional heatsink is stationary, so it doesn't get a chance to change the layering behavior of air when the moving air encounters the stationary heatsink.
Do you have this backwards? The spinning fan throws air outward, creating a low pressure vacuum at the center of the fan. So the fan draws its own air. You seem to suggest there is some force directing air toward the center that forms a pressure? I'm not sure what you mean. Maybe you are saying the fan somehow blows air into the fluid bearing to pressurize it (hydrostatic)? Or are you talking about the part of the spinning section that has the fins, facing away from the fluid bearing?
Again, are you sure about that? See the chart, and how at 1 atmosphere, the heat capacity doesn't change much even when you increase or decrease the pressure. Even increasing the pressure to 10 atmospheres, the heat capacity barely changes.
So, I think you are talking about negligible effects, even any changes would be very negligible.
But, also, can you see that the interesting effect is not only due to specific heat capacity, but other effects? More of how the air behaves/flows/layers.