you: That would be the last question.
On Hawking's black hole entropy, the issue was physical entropy. Strominger and Vafa used string theory to make the calculation (Bekenstein/Hawking). The issue of information (reduction of Shannon entropy) being lost in black holes came up separately. AFAIK, none of the physicists confused the two entropy formulations.
Also, the "ergod" is not the inverse of the void.
As before, if we are going to explore ergodic theory - we need to distinguish between physics and mathematics.
And concerning ergodic theory and singularities, the zero point is the poison pill:
Boltzmann thought of the proper average values to identify with macroscopic features as being averages over time of quantities calculable from microscopic states. He wished to identify the phase averages with such time averages. He realized that this could be done if a system started in any microscopic state eventually went through all the possible microscopic states. That this was so became known as the ergodic hypothesis. But it is provably false on topological and measure theoretic grounds. A weaker claim, that a system started in any state would go arbitrarily close to each other microscopic state is also false, and even if true would not do the job needed.
The mathematical discipline of ergodic theory developed out of these early ideas. When can a phase average be identified with a time average over infinite time? G. Birkhoff (with earlier results by J. von Neumann) showed that this would be so for all but perhaps a set of measure zero of the trajectories (in the standard measure used to define the probability function) if the set of phase points was metrically indecomposable, that is if it could not be divided into more than one piece such that each piece had measure greater than zero and such that a system started in one piece always evolved to a system in that piece.
But did a realistic model of a system ever meet the condition of metric indecomposability? What is needed to derive metric indecomposability is sufficient instability of the trajectories so that the trajectories do not form groups of non-zero measure which fail to wander sufficiently over the entire phase region. The existence of a hidden constant of motion would violate metric indecomposability. After much arduous work, culminating in that of Ya. Sinai, it was shown that some "realistic" models of systems, such as the model of a gas as "hard spheres in a box," conformed to metric indecomposability. On the other hand another result of dynamical theory, the Kolmogorov-Arnold-Moser (KAM) theorem shows that more realistic models (say of molecules interacting by means of "soft" potentials) are likely not to obey ergodicity in a strict sense. In these cases more subtle reasoning (relying on the many degrees of freedom in a system composed of a vast number of constituents) is also needed.
If ergodicity holds what can be shown? It can be shown that for all but a set of measure zero of initial points, the time average of a phase quantity over infinite time will equal its phase average. It can be shown that for any measurable region the average time the system spends in that region will be proportional to the region's size (as measured by the probability measure used in the microcanonical ensemble). A solution to a further problem is also advanced. Boltzmann knew that the standard probability distribution was invariant under time evolution given the dynamics of the systems. But how could we know that it was the only such invariant measure? With ergodicity we can show that the standard probability distribution is the only one that is so invariant, at least if we confine ourselves to probability measures that assign probability zero to every set assigned zero by the standard measure.
We have, then, a kind of "transcendental deduction" of the standard probability assigned over microscopic states in the case of equilibrium. Equilibrium is a time-unchanging state. So we demand that the probability measure by which equilibrium quantities are to be calculated be stationary in time as well. If we assume that probability measures assigning non-zero probability to sets of states assigned zero by the usual measure can be ignored, then we can show that the standard probability is the only such time invariant probability under the dynamics that drives the individual systems from one microscopic state to another.
As a full "rationale" for standard equilibrium statistical mechanics, however, much remains questionable. There is the problem that strict ergodicity is not true of realistic systems. There are many problems encountered if one tries to use the rationale as Boltzmann hoped to identify phase averages with measured quantities relying on the fact that macroscopic measurements take "long times" on a molecular scale. There are the problems introduced by the fact that all of the mathematically legitimate ergodic results are qualified by exceptions for "sets of measure zero." What is it physically that makes it legitimate to ignore a set of trajectories just because it has measure zero in the standard measure? After all, such neglect leads to catastrophically wrong predictions when there really are hidden, global constants of motion. In proving the standard measure uniquely invariant, why are we entitled to ignore probability measures that assign non-zero probabilities to sets of conditions assigned probability zero in the standard measure? After all, it was just the use of that standard measure that we were trying to justify in the first place.
In any case, equilibrium theory as an autonomous discipline is misleading. What we want, after all, is a treatment of equilibrium in the non-equilibrium context. We would like to understand how and why systems evolve from any initially fixed macroscopic state, taking equilibrium to be just the "end point" of such dynamic evolution. So it is to the general account of non-equilibrium we must turn if we want a fuller understanding of how this probabilistic theory is functioning in physics.
What happens BTW if some supernatural entity puts His finger on the scales and violates the law of equal a priori probabilities of degenerate states?
There are 100 billion neurons in the human brain. There are 100 billion stars in the Milky Way. One star per neuron. 25 thousand neurons can fly the F-22, one of the most complex, awesome machines we have ever created. The F-22 is an illusion, a division of reality, not natural--something imagined and only a function of a few neurons: a pattern with its own laws. There are no laws, no patterns in nature. Laws and patterns are only the workings of our ergodic imaginings. The ergod is the tao--a word, but not the word.