Could collision of two black holes, each the mass of the moon, be a good probe for quantum gravity? This is a naive question (to say the least), so I’m expecting another tsunami of downvotes :)
I’ve been learning about the “ultraviolet catastrophe”, and Planck’s solution to it, which theorizes that blackbody radiation doesn’t not grow exponentially with frequency, but instead peaks around the violet wavelength and then trails off.
Violet wavelength is ~ 450 nanometers.
Then, I started wondering, would a similar theory be valid for gravitational waves?
In other words, would we expect gravitons to behave like photons with respect to their radiation spectra?
My initial suspicion is that no, gravitational wave radiation will not behave like blackbody radiation, but figured I would ask anyway.
For a concrete example:
For 2 black holes of Mass the size of our sun, the smallest gravitational wave wavelength is a few kilometers (presumably close to the Schwarzchild radius of the black holes).
However, if you took 2 black holes the size of a small moon (about 10^19 kg), their Schwartzschild radius would be 14 nanometers.
Wolfram Alpha calculation
Presumably they would be radiating gravitational waves at a wavelength < 100 nanometers in the final moments before merging.
(If gravitational waves were to behave like blackbody radiation, then we would expect more gravitons to be emitted at lower frequencies than at the higher frequencies.)
(I have similar questions about gluons and muons, but that would be for another time)
 A: To answer your overall question: yes, thermally gravitons are expected to behave (in a first, exceedingly accurate approximation) just like photons. This is because gravitons, as much as photons, are massless bosons, and all the massless bosons (provided that they can be treated approximately as non-interacting particles) possess the same blackbody spectrum. Moreover, $E=\hbar\omega$ holds for any kind of massless particle (be it a boson or a fermion).
Just to be clear, when we speak of gravitons we usually interpret these as weak perturbations of the gravitational field. As such, all the complications due to the strong non-linearity of the Einstein equations do not arise in the context of gravitons, just by definition.
However, I must point out that several (more or less explicit) assumptions in your question were wrong, and this leads to much confusion in answering. First of all no, blackbody radiation does not peak around the wavelength of violet. This is because the peak of the blackbody spectrum actually depends on temperature through Wien's law,
$$
\lambda_{\text{peak}}=\frac{2.898 \cdot 10^{-3}\ \text{m}\cdot\text{K}}{T}
$$
Therefore the higher the temperature, the shorter the wavelength of the peak. If the temperature is sufficiently low then the peak will occur in the infrared or radio portion of the spectrum, while if the temperature is sufficiently high the peak will occur well beyond the visible (and indeed its wavelength can be arbitrarily short at increasing temperature).
Second of all, colliding black holes (or neutron stars, or whatever) are not expected to emit gravitational radiation with the blackbody spectrum. Indeed, the spectrum of the gravitational radiation emitted in a collision depends on the kinematic of the collision (mainly the masses of the bodies and their relative angular momentum), with no reference, for example, on temperature. So no, quantum effects cannot be explored in a collision through the mechanism of blackbody radiation, since there is not any in the sense that I believe you are implying. Nonetheless, it is believed by some that quantum effects may be detectable through the radiation spectrum of a collision.
On the other hand, black holes are indeed expected to emit radiation with the blackbody spectrum, in the form of Hawking radiation. Here the actual, thermal temperature is replaced by Hawking's temperature
$$
T_{H}=\frac{\hbar c^{3}}{8\pi G \kappa_{B}M}
$$
which is just a measure of the inverse mass $M$ of the black hole. In particular, since the power of the emitted radiation goes like $T_{H}^{4}\propto M^{-4}$, this kind of radiation is very faint, unless the black hole is quite small. However, this kind of radiation is not gravitonic: all kind of particles are emitted in the Hawing process. Hawking radiation is a strong candidate phenomenon for the study of quantum gravity.
Some sidenotes. For massive particles one has $E=\sqrt{m^{2}c^{4}+p^{2}c^{2}}=\sqrt{m^{2}c^{4}+h^{2}c^{2}/\lambda^{2}}$ to replace the formula $E=hc/\lambda$, which is valid in the limit $m\to 0$. Muons do not behave like photons, first of all because they are massive, second of all because they are fermions and last of all because they are unstable particles. Gluons also do not behave like photons, but this time this is due to their interactions: at low temperatures they are confined and cannot be detected (in particular they do not have a blackbody spectrum), and they become unconfined and approximately non-interacting at very high temperatures, in conditions which are not easy to find in our visible universe (perhaps in the inner core of neutron stars, or early on in the very first instants after the Big Bang).
