Intuitive reason for the $T^4$ term in Stefan Boltzmann law The Stefan Boltzmann Law gives a relation between the total energy radiated per unit area and the temperature of a blackbody. Specifically it states that, $$ j= \sigma {T}^4$$ Now using the thermodynamic derivation of the energy radiated we can derive the above relation, which leads to $T^4$. But is there any intuitive reason for the $T^4$ term? 
 A: There's roughly $kT$ energy in each active mode. The active modes are characterized by momenta which live inside a sphere of radius proportional to $kT$, which has volume proportional to $T^3$. Multiplying these factors gives $T^4$, and the result clearly generalizes to $T^{d+1}$ in general dimension.
A: If you know Quantum Mechanics, you  know that you can set length to have dimensions of the inverse of energy. This means that $j$ must have dimensions of energy to the four. If you consider that the only variable with energy units is the temperature, then  the energy density must be proportional to $T^4$.
If you consider additional constants with energy dimensions, like a mass, then the derivation is no longer valid.
A: The most basic argument you can make is dimensional.  You have three constants: $k$ (energy per Kelvin), $h$ (energy per Hertz),  as $c$ (meters per second), and one parameter, $T$ (Kelvin), and need to come up with power per unit area, or energy per second per meter-squared:
Dimensionally:
Energy $\rightarrow E = kT$
Time $\rightarrow h/E = \frac{h}{kT}$, which gives:
Power = Energy / Time $\rightarrow  \frac{k^2T^2}{h}$
Length = speed $\times$ Time $\rightarrow c\frac{h}{kT}$
Area = $\frac{c^2h^2}{k^2T^2}$
Combine that into Power per unit Area:
$ j \propto \frac{k^4}{h^3c^2}T^4 $.
A: Classical radiation energy density is infinite, Rayleigh-Jeans law. How can the thermodynamic derivation, which relays on the relation
$$ P=\frac{E_{dens}}{3} , $$
 be right? If it works ignoring RJ law it means that the thermodynamic relations used here are more general than classical physics.
It looks like that the only needed assumption is that a $P=P(T)$ finite relationship exists which is what Boltzmann must have assumed (before Plank's time) thanks to an article by Adolfo Bartoli, who "demonstrated" that radiation has a pressure. This article is cited in Boltzmann paper, he could read and speak italian as a good austrian. In conclusion the SB law can't be derived classically; it was just derived before QM.
A: You asked for intuitive.  If you are going to change the direction and hence the momentum of a point mass, you need kinetic energy; $E = \frac{1}{2} m v^{2}$, since you just have the one direction to worry about.  The derivative, the change, yields momentum, mv. The real world is in 3 dimensions.  Integrating, the reverse of derivation, means you need the $4$ as an exponent, since the point of the equation the energy (in watts typically) you need to change the temperature.
