How to derive the temperature dependence of internal energy of a photon gas from general principles

Given a photon gas, one can use standard statistical mechanics methods to obtain the energy
$$U=aVT^4,$$
with $a$ a messy constant whose precise details do not matter. This is a fairly involved process, but it is doable. However, my question is, can one obtain this temperature dependence from more roundabout methods? To be more precise, it is much easier to show that the pressure $p$ is related to volume $V$ and internal energy $U$ by
$$p=frac{U}{3V},$$
and that the energy density is independent of the volume
$$left(frac{partial U}{partial V}right)_T=frac{U}{V}.$$
Is it possible to use just these 2 expressions to obtain
$$Upropto VT^4$$
or is it necessary to go to the full statistical machinery to derive the volume and temperature dependence of the internal energy?