Heat capacity from dispersion relation

[dark_matter_is_neat]

Homework Statement

At low temperatures, a 2-d d-wave superconductor can be described as a gas of non-interacting fermions that follow dispersion relation $E(k) = \sqrt{a^{2}k_{x}^{2}+b^{2}k_{y}^{2}}$ where a and b are positive constants. The fermion number is not conserved. Determine how the specific heat of the system depends on temperature.

Relevant Equations

$U = \int_{0}^{\infty} D(E)*E*\frac{1}{e^{\frac{E}{k_{B}T}} + 1} dE$
$C = \frac{dU}{dT}$

For me the part of the problem that is giving me issues is obtaining the density of states, since typically how you would calculate D(E) as D(E) = $\frac{A}{2 \pi} *k*\frac{dk}{dE}$ but this shouldn't work since this assumes angular symmetry in k space which this dispersion relationship doesn't have. This dispersion relation essentially makes an ellipse in k space so the density of states should be $D(E) = \frac{A}{4 \pi^{2}} * \frac{dk_{x}dk_{y}}{dE}$ which I'm not really sure how to calculate. Once I get the density of states it should be pretty trivial to get the temperature dependence of the heat capacity, since after appropriate usage of u substitution the expression for U will work out to just be a constant times some power of T.