  Holographic dark energy (HDE) models, underlain by an effective quantum field
theory (QFT) with a manifest UV/IR connection, have become a convincing
candidate for the dark energy in the universe. On the other hand, the maximum
number of quantum states a conventional QFT in the box of size $L$ is capable
to describe, refer to those boxes which are on the brink of experiencing a
sudden collapse to a black hole. Another restriction on the underlying QFT is
that the UV cutoff, which cannot be chosen independently of the IR cutoff and
therefore becomes a function of time in a cosmological setting, should stay the
largest energy scale even in the standard cosmological epochs preceding a dark
energy dominated one. We show that, irrespective of whether one deals with the
saturated form of HDE or takes a certain degree of non-saturation in the past,
the above restrictions cannot be met in a radiation-dominated universe, an
epoch in the history of the universe which is expected to be perfectly
describable within conventional QFT.
