  The holographic cut-off, in the formalism of generalized holographic dark
energy (HDE), is generalized to depend on $L_\mathrm{IR} = L_\mathrm{IR} \left(
L_\mathrm{p}, \dot L_\mathrm{p}, \ddot L_\mathrm{p}, \cdots, L_\mathrm{f}, \dot
L_\mathrm{f}, \cdots, a\right)$, where $L_\mathrm{p}$ and $L_\mathrm{f}$ are
the particle horizon and future horizon respectively, and $a$ is the scale
factor of the universe. Based on such formalism, we showed that the Barrow
entropic dark energy (DE) model is equivalent to the generalized HDE where the
respective holographic cut-off is determined by two ways -- (1) in terms of
particle horizon and its derivative and (2) in terms of future horizon and its
derivative. Interestingly, such cut-off turns out to depend up-to first order
derivative of $L_\mathrm{p}$ or $L_\mathrm{f}$ respectively. Such equivalence
between the Barrow entropic dark energy and the generalized HDE is extended to
the scenario where the exponent of the Barrow entropy allows to vary with the
cosmological expansion of the universe. In both the cases (whether the Barrow
exponent is a constant or varies with the cosmological evolution), we determine
effective equation of state (EoS) parameter from the generalized holographic
point of view, which, by comparing with the Barrow DE EoS parameter, further
ensures the equivalence between the Barrow entropic dark energy and the
generalized HDE.
