Gravitons are bosons. As such they should obey the Bose-Einstein statistics at low temperature. And this means they should form a Bose-Einstein condensate at temperature close to absolute zero. However I imagine a condensate of gravitons to be a systems which generates a very strong gravitational field, because of the maximally dense packing of gravitons in such a condensate. Could it therefor be that a black hole is in fact a Bose-Einstein condensate of gravitons?
As a start to an answer, the formula for the temperature of the interior of a black hole is known. In particular:
For small black holes, we study their black body radiation and see so little emission that the temperature is about 1/10,000,000 of a degree above absolute zero. Larger black holes would be even colder because they let less radiation escape. That means black holes are colder than space itself (about 2.7 degrees above absolute zero).
But, the harder question is whether a black hole would transform the ordinary matter, dark matter (if such thing exists), and photons that fall into it into gravitons.
The average density of a black hole, measured as mass divided by volume within the event horizon, is never more than slightly more dense (a few percent or so) than a neutron star which keeps the lion's share of its mass in the form of ordinary matter (i.e. neutrons)1, rather than transforming any substantial share of its mass into gravitons. And, in the absence of better and experimentally validated models of the structure of the interior of a black hole (which may be not just practically, but theoretically impossible), there is no way to test the composition of the interior of a black hole and no really compelling reason for it not to be made of really cold ordinary matter.
One can imagine a black hole composed of a Bose-Einstein condensate, as the two articles cited in the comments to the question do. Dvali (2012) argues that a work around to the entropy problem that I identify below can make this possible:
We reformulate the quantum black hole portrait in the language of modern condensed matter physics. We show that black holes can be understood as a graviton Bose-Einstein condensate at the critical point of a quantum phase transition, identical to what has been observed in systems of cold atoms. The Bogoliubov modes that become degenerate and nearly gapless at this point are the holographic quantum degrees of freedom responsible for the black hole entropy and the information storage. They have no (semi)classical counterparts and become inaccessible in this limit. These findings indicate a deep connection between the seemingly remote systems and suggest a new quantum foundation of holography. They also open an intriguing possibility of simulating black hole information processing in table-top labs.
But, as far as I know, this article's hypothesis about entropy is not widely accepted. The cited follow up article to Dvali (2012), which is Alfaro (2016), seems to be a bit more carefully reasoned. Alfaro's abstract states:
We analyze in detail a previous proposal by Dvali and Gomez that black holes could be treated as consisting of a Bose-Einstein condensate of gravitons. In order to do so we extend the Einstein-Hilbert action with a chemical potential-like term, thus placing ourselves in a grand-canonical ensemble. The form and characteristics of this chemical potential-like piece are discussed in some detail. After this, we proceed to expand the ensuing equations of motion up to second order around the classical Schwarzschild metric so that some non-linear terms in the metric fluctuation are kept. We argue that the resulting equations could be interpreted as the Gross-Pitaevskii equation describing a graviton Bose-Einstein condensate trapped by the black hole gravitational field. Next we search for solutions and, modulo some very plausible assumptions, we find out that the condensate vanishes outside the horizon but is non-zero in its interior. Based on hints from a numerical integration of the equations we formulate an ansatz and eventually find an exact non-trivial solution for a mean-field wave-function describing the graviton Bose-Einstein condensate in the black hole interior. Based on this we can rederive some of the relations involving the number of gravitons N and the black hole characteristics, summarized in its Schwarzschild radius, along the lines suggested by Dvali and Gomez. These relations are parametrized by a single parameter —a dimensionless chemical potential.
It is less clear in the follow up article that the graviton BEC must be the sole content of the black hole and neither paper addresses any process by which ordinary matter sucked into a black hole is transformed into part of a graviton BEC.
I am not aware of any known process that would convert ordinary matter to gravitons in an ordinary black hole and I am not aware of research that really singles out this possibility. Indeed, a conversion of ordinary matter absorbed by the black hole into gravitons would violate baryon number and lepton number conservation and so this interaction is forbidden in the Standard Model and any reasonably plausible quantum gravity theory that preserves B and L conservation. (Could the graviton be the Goldstone boson associated with the B and L conservation laws under Noether's Theorem? There isn't an obvious reason why these could be connected, but gravity is a force that couples to absolutely everything, and in theory, if it couples to everything it could be a bridge to convert any form of mass-energy to any other form of mass-energy.)
Similarly, a graviton BEC scenario is inconsistent with the possibility of a charged black hole (the main alternatives to Schwarzchild and Kerr black holes which are called Reissner–Nordström black holes if the angular momentum is zero and are called Kerr–Newman black holes if there is both angular momentum and electric charge), since gravitons lack electromagnetic charge.
We do know (and can calculate) the total entropy of a black hole and might be able to theoreticaly rule out some versions of the possibilites using this formula, but otherwise I don't know how you could tell the difference.
A Platonic ideal of a Bose-Einstein condensate has entropy of zero which is different from the entropy of a black hole, so naively it would seem that a black hole cannot be purely a perfect Bose-Einstein condensate of gravitons, even though there could be some bosons in a black hole which are in that state. Short of a true Platonic ideal of a "perfect Bose-Einstein condensate", however, one can seem "superfluidity" at very low temperatures, which does not have zero entropy.
It seems more plausible that the gravitons inside a black hole may be in a Bose-Einstein condensate state than that everything inside a black hole is a Bose-Einstein condensate of gravitons.
1 This is true only for black holes created by the collapse of a star and subsequent accumulation of mass. In principle, "primordial black holes" could have a greater density, but no such black holes have ever been observed and conjectures about how primordial black holes could have been created in the very early universe are purely speculative.
You will not succeed in understanding black holes by analogy with Bose condensates. Even non-interacting particles can undergo Bose condensation, but GR is a nonlinear theory, so you should think of gravitons as interacting particles, and the nonlinear interaction in black holes is very strong indeed. (Do not be confused by linearized GR, aka weak-field GR, which is only an approximation. It's good enough for understanding gravitational waves, but it does not get into interactions.)
I am no expert in this field. As far as I understand, gravitons only appear in the quantization of gravitational waves not in static solutions of the Einstein field equations. Similarly, photons appear only in quantized electromagnetic waves and not in static electric fields as exemplified by the Coulomb force.