  The Debye model of the specific heat of solid at low temperatures is
incorporate in the Entropic Gravity Theory (EGT). Rather of a smooth surface,
the holographic screen is considered as an oscillating elastic membrane, with a
continuous range of frequencies, that cuts off at a maximum (Debye)
temperature, $T_D$. We show that at low temperatures $T < T_D$, the
conservation of the equivalence principle in EGT requires a modification of the
Davies-Unruh effect. While the maintenance of Davies-Unruh effect requires a
violation of the equivalence principle. These two possibilities are
equivalents, because both can emulate the same quantity of dark matter.
However, in both cases, the central mechanism is the Davies-Unruh effect, this
seems to indicate that the modification of the Davies-Unruh effect emulates
dark matter which in turn can be see as a violation of the equivalence
principle. This scenario is promising to explain why MOND theory works at very
low temperatures (accelerations) regime, i. e., the galaxies sector. We also
show that in the intermediate region, for temperatures slightly lower or
slightly higher than Debye temperature, EGT predicts the mass-temperature
relation of hot X-ray galaxy clusters.
