Heat Capacity of Blackbody 2?

To add to the mystery of the concept of a blackbody as a "cavity absorbing all incident radiation", we now ask about the heat capacity/specific heat of such a thing?

We know that the heat energy/internal energy per unit time (radiance) radiated from a blackbody scales with temperature T like T^4, but what is the connection between T and absorbed/radiated heat energy Q(T), that is what is the heat capacity dQ/dT of a blackbody?

There are different answers:

1. If the cavity is filled with a perfect gas, then Q scales like T.

2. If the cavity is filled with a solid, then Q should scale like T^2 according to Mathematical Physics of Blackbody Radiation.

3. If the cavity filled with a photon gas, then Q scales like T^4.

We thus get a heat capacity dQ/dT scaling with temperature in a wide range from T^0 = constant to T^3.

This is yet another indication that the concept of a blackbody is confusing since it is not at all clear what physics it models, and thus is unfortunate as a scientific concept.