  We formulate power-law holographic dark energy, which is a modified
holographic dark energy model based on the extended entropy relation arising
from the consideration of state mixing between the ground and the excited ones
in the calculation of the entanglement entropy. We construct two cases of the
scenario, imposing the usual future event horizon choice, as well as the Hubble
one. Thus, the former model is a one-parameter extension of standard
holographic dark energy, recovering it in the limit where power-law extended
entropy recovers Bekenstein-Hawking one, while the latter belongs to the class
of running vacuum models, a feature that may reveal the connection between
holography and the renormalization group running. For both models we extract
the differential equation that determines the evolution of the dark-energy
density parameter and we provide the expression for the corresponding
equation-of-state parameter. We find that the scenario can describe the
sequence of epochs in the Universe evolution, namely the domination of matter
followed by the domination of dark energy. Moreover, the dark-energy equation
of state presents a rich behavior, lying in the quintessence regime or passing
into the phantom one too, depending on the values of the two model parameters,
a behavior that is richer than the one of standard holographic dark energy.
