Why $C_p$ is zero for photon gas?

For photon gas, We know that
$$U=u(T)V, p =frac{u(T)}{3}$$
As $$G=U+pV-TS=uV+frac{u}{3}V-frac{4u}{3}V=0$$
$$dG=-SdT+Vdp=0$$
$$Rightarrow s=frac{dp}{dT}not=0$$
$$dp=0Rightarrow dT=0$$
which means that $C_p=infty$.

which is obvious as pressure is a function of energy density which is a function of temperature and thus the pressure is a function of temperature. So I can’t vary one of them keeping another fix.

But Why? I’m able to understand it on a thermodynamic level but not at the microscopic level.

So What’s the microscopic point of view explanation for $C_p$ to be infinite for ideal gas?