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Taking into account a fractal structure for the black hole horizon, Barrowargued that the area law of entropy is modified due to quantum-gravitationaleffects (Barrow in Phys Lett B 808:135643,https://doi.org/10.1016/j.physletb.2020.135643, 2020). Accordingly, thecorrected entropy takes the form $S \sim A^{1+\mathrm{\Delta}/2}$, where$0\leq\mathrm{\Delta}\leq1$, indicates the amount of the quantum-gravitationaldeformation effects. In this paper, based on Barrow entropy, we first derivethe modified gravitational field equations through the Clausius relation. Wethen consider the Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric as thebackground metric and derive the modified Friedmann equations inspired byBarrow entropy. In order to explore observational constraints on the modifiedBarrow 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 Barrowcosmology with a phantom dark energy compared to $\mathrm{\Lambda}$CDM, sotensions between low redshift determinations of the Hubble constant and cosmicmicrowave background (CMB) results are slightly reduced. On the other hand, incase of dataset "Planck + Planck-SZ + CFHTLenS + Pantheon + BAO + BAORSD" thereis a slight amelioration in $\sigma_8$ tension in Barrow cosmology with aquintessential dark energy compared to the standard model of cosmology.Additionally, for a more reliable comparison, we also constrain the wCDM modelwith the same datasets, where our results exhibit a satisfying compatibilitybetween Barrow cosmology and wCDM.

Observational constraints of the modified cosmology through Barrow entropy