So say there's a cube in space acting as a blackbody.
Each side is 2 metres. Initial cube temperature is 400 Kelvin. Mass is 15 kg.
Say the heat capacity is 500 J/kgK. How would that affect thermal radiation? Or is it not a factor? How does it affect the rate of cooling? And how would you calculate this?
We can answer your question by considering the first law of thermodynamics on a body with uniform temperature $T$, which is only losing thermal energy via radiation:
$$ \frac{dU}{dt}=\dot{Q} - \dot{W}$$
where
$$ U = \rho c_p T = \frac{m}{V}c_pT$$
$$ \dot {Q} = -\epsilon \sigma T ^4$$
$$ \dot{W} =0 $$
where $ \dot {Q} = -\epsilon \sigma T ^4$ is the emitted radiation which depends on the emissivity $\epsilon$.
Substituting into the first law,
$$ \frac{m}{V}c_p \frac{dT}{dt} = -\epsilon \sigma T ^4$$
Integrate from $t=0$ to $t$, noting that $T(0) = T_0$, and solve for $t$:
$$ t = \frac{3 m c_p}{V \epsilon \sigma}\left(T^{-1/3} - T_0^{-1/3}\right)$$
You can see that heat capacity definitely affects how fast the body cools. Specifically, larger heat capacity means longer cooling times since more energy is stored in the body at a given temperature.
The temperature determines the rate at which energy is radiated away (other factors, such as the emissivity, the temperature of the surroundings, and the geometry are also relevant).
The heat capacity determines how quickly the temperature changes as internal energy is changed. All other things being equal, a hot object with a high heat capacity will cool more slowly than one with low heat capacity because it must radiate away more energy to decrease its temperature.
