Fine-grained state counting for black holes in loop quantum gravity
  A state of a black hole in loop quantum gravity is given by a distribution of spins on punctures on the horizon. The distribution is of the Boltzmann type, with the area playing the role of the energy. In investigations where the total area was kept approximately constant, there was a kind of thermal equilibrium between the spins which have the same analogue temperature and the entropy was proportional to the area. If the area is precisely fixed, however, multiple constraints appear, different spins have different analogue temperatures and the entropy is not strictly linear in the area, but is bounded by a linear rise. 
A. Ghosh and P. Mitra