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We show that in the standard derivation of the holographic dark energy someconceptual issues are present. In particular, the formula used in theliterature to avoid black holes is not suitable in expanding universes. A moresuitable expression for a holographic motivated dark energy must contain theenergy density of the remaining matter content of the universe. However, weshow that under some reasonable hypothesis, we can obtain a new physicallymotivated expression for the holographic dark energy. By considering anappropriate time-dependence of the saturation-level parameter, a de Sitterphase arises. Moreover, by adopting an argument similar to the originalBekenstein one, our approach justifies a correction of the Bekenstein-Hawkingentropy for non-static isotropic expanding universes. Finally, we write downthe equation of state of a black hole embedded in Friedmann spacetimes.

New holographic dark energy and the modified Bekenstein-Hawking entropy