Rate of spontaneous emission and energy conservation

The english wikipedia article https://en.wikipedia.org/wiki/Spontaneous_emission#Rate_of_spontaneous_emission (26.08.2020, 16:45) gives the rate for spontaneous emission as
$$
Gamma_{rad}(omega) = frac{omega^3n|mu_{12}|^2}{3pi varepsilon_0 hbar c^3}
$$

Isn’t this missing some sort of condition for omega to ensure energy conservation like $delta(omega – omega_{12} )$ ? This formula makes it look as if spontaneous emission always has the form of $f(omega) = k omega^3$, where $k$ is some constant.