  A study on the effects of implementing the Granda-Oliveros infrared cutoff in
the recently introduced Barrow Holographic Dark Energy model is presented, and
its cosmological evolution is investigated. We find how the deformation
parameter, $\Delta$, affects the values of $H(z)$, and find that from this
model it is possible to obtain an accelerated expansion regime of the universe
at late times. We also obtain that increasing $\Delta$ causes the EoS parameter
to transition from quintessence to phantom. In addition, we show that the model
can be used to describe the know eras of dominance. Finally, after studying the
stability of the proposed model, a fit of the corresponding parameters is
preformed, utilizing the measurements of the expansion rate of the universe,
$H(z)$. The best fit of the parameters is found to be $(\alpha,\, \beta,\,
\Delta) =
(1.00^{+0.02}_{-0.02},\,0.69^{+0.03}_{-0.02},\,0.000^{+0.004}_{-0.000})$ at
$1\sigma$ C.L, for which the Bekenstein-Hawking relation is favored.
