  An often fruitful route to study quantum gravity is the determination and
study of quantum mechanical models--that is, models with finite degrees of
freedom--that capture the dynamics of a black hole's microstates. An example of
such a model is the superconformal quantum mechanics of Yang-Mills instantons,
which has a proposed gravitational dual description as M-theory on a background
of the form $X_7\times S^4$. This model arises in the strongly-coupled limit of
the BFSS matrix model with additional fundamental hypermultiplets, offering a
route towards useful numerical simulation. We construct a six-parameter black
hole solution in this theory, which is generically non-supersymmetric and
non-extremal, and is shown to arise in an "ultra-spinning" limit of the
recently-found six-parameter AdS$_7$ solution. We compute its thermodynamic
properties, and show that in the supersymmetric limit the entropy and on-shell
action match precisely the expected results as computed from the superconformal
index of the quantum mechanics, to leading order in the supergravity regime.
The low-lying spectrum thus provides access to the dynamics of near-extremal
black holes, whose spectra are expected to receive strong quantum corrections.
