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.