  The thermodynamics-gravity conjecture reveals that one can derive the
gravitational field equations by using the first law of thermodynamics and vice
versa. Considering the entropy associated with the horizon in the form of
non-extensive Tsallis entropy, $S\sim A^{\beta}$ here we first derive the
corresponding gravitational field equations by applying the Clausius relation
$\delta Q=T \delta S$ to the horizon. We then construct the Friedmann equations
of Friedmann-Lema\^itre-Robertson-Walker (FLRW) universe based on Tsallis
modified gravity (TMG). Moreover, in order to constrain the cosmological
parameters of TMG model, we use observational data, including Planck cosmic
microwave background (CMB), weak lensing, supernovae, baryon acoustic
oscillations (BAO), and redshift-space distortions (RSD) data. Numerical
results indicate that TMG model with a quintessential dark energy is more
compatible with the low redshift measurements of large scale structures by
predicting a lower value for the structure growth parameter $\sigma_8$ with
respect to $\Lambda$CDM model. This implies that TMG model would slightly
alleviate the $\sigma_8$ tension.
