@dmckee sorry I don't quite follow your second sentence, doesn't deep inelastic scattering contradict what you're saying (I may well be confusing myself here)

is there any meaningful connection between K and the Ricci tensor? or is it overly complicated to use in the FRW metric?

I'm not sure how useful it is to talk about an infinite dust cloud as you couldn't say that I would collapse to a star if there were some small inhomogeneity as it is infinite in extent. Also since 0K cant be achieved, the dust particles will have some kinetic energy so you get random fluctuations in density as the move around, interact and collide, eventually there will be less and less homogeneity and eventually stars I guess.

Ahh okay, so we set it to zero for $\epsilon$ below zero but that doesn't matter anyway as we can neglect any terms which don't depend on T as that's all we're interested in for the heat capacity. I get out $C/N=\frac{(\pi{k_B})^2N}{6\epsilon_F}T$ which gives the total heat capacity of a metal the required form of $AT^3+BT$ (I am kind of assuming this dependence on T is general for d dimensions)

Low T limit, couple of things confusing me if you could take a quick look below

Okay, after looking at it for quite a while I'm pretty happy bar a couple of things, firstly the function $f(\epsilon)$ should vanish for epsilon goes to minus infinity, however in this case $f(\epsilon)=g(\epsilon)\epsilon$ and the density of sates is just a constant, so f diverges at minus infinity? Also, I'm confused why the integral's upper limit can be changed to $\mu$ (or even to $\epsilon_F$ as I've seen in some places). Is it that the fermi function goes quickly to zero within a couple of $k_BT$ of the Fermi energy (or chemical potential) and T is small so limit is pretty much $\mu$