  Taking into account a fractal structure for the black hole horizon, Barrow
argued that the area law of entropy is modified due to quantum-gravitational
effects (Barrow in Phys Lett B 808:135643,
https://doi.org/10.1016/j.physletb.2020.135643, 2020). Accordingly, the
corrected entropy takes the form $S \sim A^{1+\mathrm{\Delta}/2}$, where
$0\leq\mathrm{\Delta}\leq1$, indicates the amount of the quantum-gravitational
deformation effects. In this paper, based on Barrow entropy, we first derive
the modified gravitational field equations through the Clausius relation. We
then consider the Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric as the
background metric and derive the modified Friedmann equations inspired by
Barrow entropy. In order to explore observational constraints on the modified
Barrow cosmology, we employ two different combinations of available datasets,
mainly "Planck + Pantheon + BAO" and "Planck + Planck-SZ + CFHTLenS + Pantheon
+ BAO + BAORSD" datasets,. According to numerical results, we observe that the
"Planck + Pantheon + BAO" dataset predicts higher values of $H_0$ in Barrow
cosmology with a phantom dark energy compared to $\mathrm{\Lambda}$CDM, so
tensions between low redshift determinations of the Hubble constant and cosmic
microwave background (CMB) results are slightly reduced. On the other hand, in
case of dataset "Planck + Planck-SZ + CFHTLenS + Pantheon + BAO + BAORSD" there
is a slight amelioration in $\sigma_8$ tension in Barrow cosmology with a
quintessential dark energy compared to the standard model of cosmology.
Additionally, for a more reliable comparison, we also constrain the wCDM model
with the same datasets, where our results exhibit a satisfying compatibility
between Barrow cosmology and wCDM.
