This research addresses extensive and non-extensive thermodynamics.
A comparison between the entropy for both different statistics are presented. The non-extensive parameter, entropic index q, is discussed. We
attempt to explore the limit of the non-extensive parameter by comparing
the theoretical results with lattice and the available experimental results.
The two thermal parameters T, B are calculated with the freeze-out condition S/T^3 = 7 for different q. The motivation of this research comes
from recent non-extensive statistics studies which showed that this standard thermodynamics failed to reproduce the freeze-out parameters. As an
application, the black-hole entropy is calculated in the quantum Generalized Uncertainty Principle (GUP) modification form. Black-hole entropy
may reveal information about the thermodynamics it belongs. This discrimination is essential to quantify the entropy in the hadron production
evolution stage and in the black-hole thermodynamics. It is concluded that
lattice QCD reproduces the extensive thermodynamics very well. Also, the
black hole appears as an extensive system.