We present how Tsallis cosmology can alleviate both $𝐻_0$ and $\sigma_8$ tensions simultaneously. Such
a modified cosmological scenario is obtained by the application of the gravity-thermodynamics
conjecture, but using the non-additive Tsallis entropy, instead of the standard Bekenstein-Hawking
one. Hence, one obtains modified Friedmann equations, with extra terms that depend on the new
Tsallis exponent that quantifies the departure from standard entropy. We show that for particular
choices we can obtain a phantom effective dark energy, which is known to be one of the sufficient
mechanisms that can alleviate $𝐻_0$ tension. Additionally, for the same parameter choice we obtain
an increased friction term and an effective Newton’s constant smaller than the usual one, and thus
the $\sigma_8$ tension is also solved. These features act as a significant advantage of Tsallis modified
cosmology