How does the power emitted during thermal radiation depend on time (if it does)? What are some sources refering to the particular relation between the power emitted during thermal radiation and time? How would the plot look like?
What are the fundamental underlying principles governing the derivation of an expression regarding the power under consideration?
I assume that by thermal radiation one means here black body radiation (but see also discussion: Black body vs. Thermal radiation). There are two things that can be meant here, depending on the context:
Black body radiation as the equilibrium state of a photon gas (or as an equilibrium between a black body and the radiation). Since we are talking about equilibrium here, we imply that there are no changes in time - as is always done in equilibrium statistical physics. We could calculation the level of fluctuations (e.g., of the radiation intensity at a given frequency), but these are also stationary fluctuations, i.e., their characteristics do not change in time.
Hot object described by a black body spectrum a hot object, like a heated piece of metal or a star is often described by Planck spectrum. Such an object is not truly in equilibrium with its environment - the radiation escapes from it and does not return. However, as most of the radiation remains confined within the object, it can be characterized by the black body spectrum with the shape determined by the temperature of the object. What happens then depends on the nature of the object: a hot piece of metal is likely to cool, a star would continue to shine in the same spectrum for a long time due to the internal sources of energy, burning fire represents a somewhat intermediate case - it burns till the fuel is exhausted.
See also:
Does fire emit black-body radiation?
How does radiation become black-body radiation?
How is light emitted by an incandescent lamp?
How a piece of metal put in the oven reaching a constant temperature of 3000 K fit the model of a hole in a cavity?
How does the power emitted during thermal radiation depend on time (if it does)? What are some sources referring to the particular relation between the power emitted during thermal radiation and time? How would the plot look like?
What are the fundamental underlying principles governing the derivation of an expression regarding the power under consideration?
The power emitted by a "black body" (or a "gray body") at constant temperature $T$ is given by the Stefan-Boltzmann law: $$ P_{\text emit} = \alpha T^4\;, $$ where $\alpha$ is a constant.
In the derivation of the above law, the temperature is assumed constant. This can be taken to mean that the radiating body is so large that the loss of energy due to radiation does not affect its temperature.
However, you want to understand how the temperature changes with time. In that case, it may be reasonable to assume the body undergoes radiative cooling and transfers heat to its environment, and thus cools (assuming the specific heat is positive, which is not necessarily the case on an astronomical scale).
The black or grey body not only emits thermal radiation, but it can also absorb thermal radiation from its environment. So, the total change in energy of the body can be approximated as: $$ \frac{\delta E}{\delta t} = P = P_{\text emit}- P_{\text absorb} = \alpha (T^4 - T_{\text env}^4)\;, $$
If we next assume that it is appropriate to use a constant specific heat, we can re-write this equation as: $$ \frac{\delta E}{\delta T}\frac{\delta T}{\delta t} \approx cM\frac{\delta T}{\delta t} = \alpha (T^4 - T_{\text env}^4)\;, $$ where $c$ is the specific heat and $M$ is the mass of the body.
If we assume the mass and the specific heat remain constant we can rewrite this as: $$ \frac{\delta T}{\delta t}\approx \beta (T^4 - T_{\text env}^4)\;,\tag{1} $$ where $\beta$ is a different constant (and is not related to the usual use of the symbol $\beta$ in statistical mechanics).
Please note that this derivation of Eq. (1) relied on many different assumptions, any of which may fail to be true in practice.
A further assumption that has not yet been stated explicitly is that the heat transfer is assumed to be entirely radiative. If conductive heat transfer is allowed then the rate of change of the temperature with time will have a different form, approximated by Newton's Law of Cooling.
Thermal radiation has an implicit dependence on time. The general expression for radiative power is the Stefan-Boltzmann Law:
$$P = \sigma AT^4$$ Where $\sigma$ is the Stefan-Boltzmann constant, $A$ is the area of the surface, and $T$ is the temperature
Any book or video detailing an introduction to modern physics or thermodynamics should cover this topic as it is a cornerstone for the beginnings of quantum mechanics and how we understand the nature of light.
Given a fixed temperature $T$ and a area $A$ this will be constant. The only condition that one could change the radiative power over time is by varying the temperature of the object over time. The plot for power over time would be different depending on the function $T(t)$ and it would be straight line for a for a fixed $T$.
It depends on what you are referring to when discussing 'Fundamental principles'. The principles behind black body radiation are generally not intuitive and it can't be derived using classical methods. If you are using 'fundamental' in the context of building it from the basis of what we know about each field in physics, you can integrate Plank's Law over the frequency $\nu$ where planks law is:
$$u(\nu ,T)=\frac{2h{\nu}^3}{c^2}\frac{1}{e^{h\nu/k_{B}T}-1}$$
To derive it from first principles requires you to derive Plank's Law which starts with considering a standing electromagnetic magnetic wave in a cavity. From there you do a lot of mathematical hoop jumps and then implement wave-particle duality as concept with an average energy. You can eventually arrive at Plank's Law and then from there do the integration.
