  In the present work, we have analyzed the behaviors of extension of
generalized Barrow holographic dark energy(`BHDE'). A ``generalized BHDE model
based on the particle and the future horizon using infrared cut-off" was
proposed by Nojiri et al. (2022). In this work, we have reviewed the
generalized BHDE extension under the assumption of a generalized HDE cut-off.
Using a scale factor of the form $a = k t^m$, the dynamics of the cosmos have
been discussed through graphic demonstration. By applying the ``open-source
emcee Python package", the values of the free parameters $k$ and $m$ are
estimated on 57 OHD points by the Markov Chain Monte Carlo (MCMC) technique. We
have examined the behavior of the equation of state (EoS) parameter, $(
p_{de})$, and dark energy density $(\rho_{de})$. We have also discussed the
equivalence of holographic dark energy (DE) with the Barrow entropic DE and its
extension. Also, we have explained quintessence and dilation dark energy models
in the context of Barrow entropic DE.
