Black holes with no space-time or no space-time only (metric)? It is generally accepted that a black hole can be defined as a trapped surface or region of space-time where gravity is so intense that light can not escape. Another definition is that a black hole is certain non-linear solution (soliton-like) to Einstein Field Equations for gravity (and its generalizations beyond standard gravity).
Certain quantum gravity approaches tell us that space-time is really an effective description of reality (being reality a quite abstract  mathematical notion). Thus, space-time is only an effective description below Planck temperature, since greater than then it is believed the true degrees of freedom of quantum gravity will appear. So, if this is true, how can the notion of black hole be defined if NO space-time is considered?
Indeed, quantum gravity searches will face finding  black holes without reference to any space-time (or any metric) at all if we knew what are the microstates that make up black holes...
In order to make this question more precise and perhaps addressable...For instance, what is a black hole, if any, in non-metric theories or beyond general theory of relativity? What is a black hole when torsion is present beyond a metric field? And when there is superspace non even coordinates? For instance, what is a black hole in Chern-Simons gauge theories of gravity? Do they include horizons as well?
In summary: what is a black hole in any theory (standard or beyond) of gravity? Any theory without black holes should be neglected? What is a "black hole" in a non-metric theory of gravity (e.g., torsional gravity)?
Remark: black holes, we know now that, are composite objects with entropy and temperature. They are highly entropic systems! Thus, despite we lack a theory of quantum gravity, we pressume black holes as describe by General Relativity and other theories are really "effective concepts" waiting for a further definition and/or experimental realization/study! But, what should we expect they are?
 A: In order to be a viable theory of quantum gravity, a non-metric theory of gravity needs to exhibit macroscopic/classical behaviour that approaches that of general relativity. A black hole solution in such a theory would be identified as a solution whose macroscopic behaviour replacates that of black hole solutions in general relativity.
How to specifically identify such solutions in particular theory cannot be answered in generality as it depends intimately on the details of the theory and how it reproduces general relativity in the macroscopic/classical limit.
A: 
Certain quantum gravity approaches tell us that space-time is really an effective description of reality … Can we define (quantum) black holes without reference to any space-time at all if we knew what are the microstates that make up black holes?

It is too early to say. These theories are not yet fully developed. They are not known to be self-consistent, nor are they known to be consistent with existing experimental evidence, nor have they been validated against new experiments. Quite a bit of theoretical work is needed before the question can be answered even hypothetically. And more still to have any reasonable degree of confidence that the answer has any bearing on reality.
