Alleviating both $H_0$ and $\sigma_8$ tensions in Tsallis cosmology

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.