  We explore a possible connection between two aspects of Loop Quantum Gravity
which have been extensively studied in the recent literature: the black-hole
area-entropy law and the energy-momentum dispersion relation. We observe that
the original Bekenstein argument for the area-entropy law implicitly requires
information on the energy-momentum dispersion relation. Recent results show
that in first approximation black-hole entropy in Loop Quantum Gravity depends
linearly on the area, with small correction terms which have logarithmic or
inverse-power dependence on the area. Preliminary studies of the
Loop-Quantum-Gravity dispersion relation reported some evidence of the presence
of terms that depend linearly on the Planck length, but we observe that this
possibility is excluded since it would require, for consistency, a contribution
to black-hole entropy going like the square root of the area.
