  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 $\delta$ that quantifies the departure from
standard entropy. We show that for particular $\delta$ 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.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Theory