  We have analyzed the Barrow holographic dark energy (BHDE) in the framework
of the flat FLRW Universe by considering the various estimations of Barrow
exponent $\triangle$. Here we define BHDE, by applying the usual holographic
principle at a cosmological system, for utilizing the Barrow entropy rather
than the standard Bekenstein-Hawking. To understand the recent accelerated
expansion of the universe, considering the Hubble horizon as the IR cut-off.
The cosmological parameters, especially the density parameter ($\Omega_{_D}$),
the equation of the state parameter ($\omega_{_D}$), energy density
($\rho_{_{D}}$) and the deceleration parameter($q$) are studied in this
manuscript and found the satisfactory behaviors. Moreover, we additionally
focus on the two geometric diagnostics, the statefinder $(r,s)$ and $O_{m}(z)$
to discriminant BHDE model from the $\Lambda CDM$ model. Here we determined and
plotted the trajectories of evolution for statefinder $(r, s)$, $(r,q)$ and
$O_{m}(z)$ diagnostic plane to understand the geometrical behavior of the BHDE
model by utilizing Planck 2018 observational information. Finally, we have
explored the new Barrow exponent $\triangle$, which strongly affects the dark
energy equation of state that can lead it to lie in the quintessence regime,
phantom regime, and exhibits the phantom-divide line during the cosmological
evolution.
