Non-existence of Extended Holographic Dark Energy with Hubble Horizon

The extended holographic dark energy model with the Hubble horizon as the infrared cutoff avoids the problem of the circular reasoning of the holographic dark energy model. We show that the infrared cutoff of the extended holographic dark energy model cannot be the Hubble horizon provided that the Brans-Dicke parameter $\omega$ satisfies the experimental constraint $\omega >{10}^4$, and this is proved as a no-go theorem. The no-go theorem also applies to the case in which the dark matter interacts with the dark energy.