In recent decades, an intensive worldwide research activity is focusing both black holes and cosmos (e.g. the dark-energy phenomenon) on the basis of entropic approaches.

The Boltzmann-Gibbs-based Bekenstein-Hawking entropy S B H ∝ A / l P 2 (A≡ area; l P ≡ Planck length) systematically plays a crucial theoretical role although it has a serious drawback, namely that it violates the thermodynamic extensivity of spatially-three-dimensional systems. Still, its intriguing area dependence points out the relevance of considering the form W ( N ) ∼ μ N γ ( μ > 1 ; γ > 0 ) , W and N respectively being the total number of microscopic possibilities and the number of components; γ = 1 corresponds to standard Boltzmann-Gibbs (BG) statistical mechanics.

For this W ( N ) asymptotic behavior, we make use of the group-theoretic entropic functional S α , γ = k [ ln ⁡ Σ i = 1 W p i α 1 − α ] 1 γ ( α ∈ R ; S 1 , 1 = S B G ≡ − k ∑ i = 1 W p i ln ⁡ p i ) , first derived by P. Tempesta in Chaos 30,123119, (2020). This functional is extensive (as required by thermodynamics) and composable, ∀ ( α , γ ) .

Being extensive means that in the micro-canonical, or uniform, ensemble where all micro-state occur with the same probability, the entropy becomes proportional to N asymptotically: S ( N ) ∝ N for N → ∞ . An entropy is composable if it satisfies that the entropy S A of a system A = B × C consisting of two statistically independent parts B and C is given in a consistent way as S A = Φ ( S B , S C ) where the composition function Φ ( x , y ) is obtained from group-theory.

We further show that ( α , γ ) = ( 1 , 2 / 3 ) satisfactorily agrees with cosmological data measuring neutrinos, Big Bang nucleosynthesis and the relic abundance of cold dark matter particles, as well as dynamical and geometrical cosmological data sets.

C. Tsallis, H.J. Jensen, Extensive composable entropy for the analysis of cosmological data, Physics Letters B 861(2025) 139238.