  We study coherent states associated to a generalized uncertainty principle
(GUP). We separately analyze the cases of positive and negative deformation
parameter $\beta$, showing that the ensuing probability distribution is a
Tsallis distribution whose non-extensivity parameter $q$ is monotonically
related to $\beta$. Moreover, for $\beta <0$ (corresponding to $q<1$), we
reformulate the GUP in terms of a one-parameter class of Tsallis entropy-power
based uncertainty relations, which are again saturated by the GUP coherent
states. We argue that this combination of coherent states with Tsallis entropy
offers a natural conceptual framework allowing to study quasi-classical regime
of GUP in terms of non-extensive thermodynamics. We substantiate our claim by
discussing generalization of Verlinde's entropic force and ensuing implications
in the late-inflation epoch. Corresponding dependence of the $\beta$ parameter
on cosmological time is derived for the reheating epoch. The obtained $\beta$
is consistent with values predicted by both string-theory models and the
naturalness principle. Further salient issues, including derivation of new
$\beta$-dependent expressions for the lowest possible value of the spin and
Immirzi parameter in Loop Quantum Gravity, and connection of our proposal with
the Magueijo--Smolin doubly special relativity are also discussed. This article
provides a more extended and comprehensive treatment of our recent letter
[Phys. Rev. D 105, L121501 (2022)].
