  We consider the holographic Friedman-Robertson-Walker (hFRW) universe on the
4-dimensional membrane embedded in the 5-dimensional bulk spacetime and fit the
parameters with the observational data. In order to fully account for the
phenomenology of this scenario, we consider the models with the brane
cosmological constant and the negative bulk cosmological constant. The
contribution from the bulk is represented as the holographic dark fluid on the
membrane. We derive the universal modified Friedmann equation by including all
of these effects in both braneworld and holographic cutoff approaches. For
three specific models, namely, the pure hFRW model, the one with the brane
cosmological constant, and the one with the negative bulk cosmological
constant, we compare the model predictions with the observations. The
parameters in the considered hFRW models are constrained with observational
data. In particular, it is shown that the model with the brane cosmological
constant can fit data as well as the standard $\Lambda$CDM universe. We also
find that the $\sigma_8$ tension observed in different large-structure
experiments can be effectively relaxed in this holographic scenario.
