  Popular approaches to quantum gravity describe black hole microstates
differently and apply different statistics to count them. Since the
relationship between the approaches is not clear, this obscures the role of
statistics in calculating the black hole entropy. We address this issue by
discussing the entropy of eternal AdS black holes in dimension four and above
within the context of a midisuperspace model. We determine the black hole
eigenstates and find that they describe the quantization in half integer units
of a certain function of the Arnowitt-Deser-Misner (ADM) mass and the
cosmological constant. In the limit of a vanishing cosmological constant (the
Schwarzschild limit) the quantized function becomes the horizon area and in the
limit of a large cosmological constant it approaches the ADM mass of the black
holes. We show that in the Schwarzschild limit the area quatization leads to
the Bekenstein-Hawking entropy if Boltzmann statistics are employed. In the
limit of a large cosmological constant the Bekenstein-Hawking entropy can be
recovered only via Bose statistics. The two limits are separated by a first
order phase transition, which seems to suggest a shift from "particle-like"
degrees of freedom at large cosmological constant to geometric degrees of
freedom as the cosmological constant approaches zero.
