It is shown that the usual entropy argument for the Gregory-Laflamme (GL)
instability for some appropriate black strings and p-branes gives
surprising agreement up to a few percent. This may provide a strong support to
the GL's horizon fragmentation, which would produce the array of
higher-dimensional Schwarzschild-type's black holes finally. On the other hand,
another estimator for the size of the black hole end-state relative to the
compact dimension indicates a second order (i.e., smooth) phase transition for
some other appropriate compactifications and total dimension of spacetime
wherein the entropy argument is not appropriate. In this case,
Horowitz-Maeda-type's non-uniform black strings or p-branes can be the final
state of the GL instability.Comment: More emphasis on a second order phase transition. The computation
result is unchange