How does I show how the heat capacity and temperature relate for 2D graphene at low temperature?

I know that for 3D solids, heat capacity and temperature relate as C proportional to T^3. Graphene has 3 acoustic branches and one has a dispersion for which ? ∝ k^2. It’s this branch at low temperature that I’m interested in. Also, according to a Stanford research paper entitled "Thermal properties of graphene:
Fundamentals and applications by Eric Pop, Vikas Varshney, and Ajit K. Roy," it states that at low temperature, C is proportional to T but I want to show a relationship between C and T using ? ∝ k^2. I know I have to integrate dU/dT but I don’t know how to set up the integrand. Could some help me with that? I can handle the rest. It would be greatly appreciated if someone could also give some explanation on the basic concept. I remember finding heat capacities in basic physics textbooks but now dealing with modes and such, I’m getting all confused about what to do with this problem, being weak in this subject, I only have experience dealing with ideal gases as reference. Any help is appreciated.