Nonassicative cosmological solitonic R-flux deformations in gauge gravity and G. Perelman geometric flow thermodynamics

We elaborate on a model of nonassociative and noncommutative gauge gravity for the de Sitter gauge group $SO(4,1)$ embedding extensions of the affine structure group $Af(4,1)$ and the Poincar\'{e} group $ISO(3,1)$. In string theory, such nonassociative gauge gravity theories are determined by star product R-flux deformations. They are new avenues to quantum gravity and geometric and quantum information theories. We analyze physically important and geometric thermodynamic properties of new classes of generic off-diagonal cosmological solitonic solutions encoding nonassociative effective sources. Particularly, we focus on modelling by such solutions of locally anisotropic and inhomogeneous dark matter and dark energy structures generated as nonassociative solitonic hierarchies. Such accelerating cosmological evolution scenarios can't be described in the framework of the Bekenstein-Hawking thermodynamic formalism. This motivates a change in the gravitational thermodynamic paradigm by considering nonassociative and relativistic generalizations of the concept of W-entropy in the theory of Ricci flows. Finally, we compute the corresponding modified G. Perelman's thermodynamic variables and analyze the temperature-like evolution of cosmological constants determined by nonassociative cosmological flows.