  Classical and quantum entropic properties of holographic dark energy (HDE)
are considered in view of the fact that its entropy is far more restrictive
than the entropy of a black hole of the same size. In cosmological settings (in
which HDE is promoted to a plausible candidate for being the dark energy of the
universe), HDE should be viewed as a combined state composed of the event
horizon and the stuff inside the horizon. By any interaction of the subsystems,
the horizon and the interior become entangled, raising thereby a possibility
that their quantum correlations be responsible for the almost purity of the
combined state. Under this circumstances, the entanglement entropy is almost
the same for both subsystems, being also of the same order as the thermal
(coarse grained) entropy of the interior or the horizon. In the context of
thermodynamics, however, only additive coarse grained entropies matter, so we
use these entropies to test the generalized second law (GSL) of gravitational
thermodynamics in this framework. While we find that the original Li's model
passes the GSL test for a special choice of parameters, in a saturated model
with the choice for the IR cutoff in the form of the Hubble parameter, the GSL
always breaks down.
