  Recently Tsallis cosmology has been presented as a novel proposal for
alleviating both $H_0$ and $\sigma_8$ tensions. Hence a universe filled with
matter and radiation as perfect fluids and considering the Tsallis entropy is
confronted using recent cosmological measurements coming from cosmic
chronometers, type Ia supernovae, hydrogen II galaxies, quasars, and baryon
acoustic oscillations. Following a Bayesian Monte Carlo Markov Chain analysis
and combining the samples, we constrain the main characteristic parameters
$\alpha = 1.031^{+0.054}_{-0.051}$ and $\delta = 1.005^{+0.001}_{-0.001}$ where
the uncertainties are $1\sigma$ confidence level. Additionally, we estimate the
current deceleration parameter $q_0=-0.530^{+0.018}_{-0.017}$, the
deceleration-acceleration transition redshift $z_T=0.632^{+0.028}_{-0.028}$ and
the age of the universe $\tau_U = 12.637^{+0.067}_{-0.066}\,\rm{Gyrs}$ which
are in agreement with the standard cosmology ($\Lambda$CDM) within $1.5\sigma$.
Furthermore, we find that the dark energy equation of state is consistent with
both phantom and quintessence behaviors within $1\sigma$ in the past and
converging to $\Lambda$CDM in the future. Additionally, we find agreement
values of $H_0$ within $1\sigma$ with the SH0ES values when the CMB distance
priors are added to the analysis, suggesting that $H_0$ tension could be
alleviated. Finally, Tsallis cosmology is preferred over $\Lambda$CDM by the
combined data that motives further studies at the perturbation level.
