  We consider Tsallis cosmology as an approach to thermodynamic gravity and
derive the bound on the Tsallis parameter to be $\beta<2$ by using the
constraints derived from the formation of the primordial light elements,
Helium, Deuterium and Litium, from the observational data from Big Bang
Nucleosynthesis (BBN) which allows only a very tiny deviation from General
Relativity (GR). Next we consider thermal dark matter (DM) freeze-out mechanism
in Tsallis cosmological era and derive bounds on the Tsallis parameter from the
observed DM relic abundance to be $1-\beta < 10^{-5}$.
