Fractional cosmology has recently emerged, based on the formalism of fractional calculus, which modifies the integer order derivative by a fractional derivative generating changes in the Friedmann equations. The standard evolution of the cosmic species densities is modified, depending on the $\mu$ fractional parameter and the age of the Universe $t_0$. The modified Friedmann equations provide a late cosmic acceleration at the background level without introducing a dark energy component. This radical approach could be a new path to tackle problems not resolved until now in cosmology. We estimate stringent constraints on the fractional and cosmological parameters using observational Hubble data, Type Ia supernovae and joint analysis to elucidate that. According to our results, the Universe would be older than the standard estimations. Finally, we analyze whether fractional cosmology can alleviate $H_0$ tension.