  We develop an approach to the theory of relativistic geometric flows and
emergent gravity defined by entropy functionals and related statistical
thermodynamics models. Nonholonomic deformations of G. Perelman's functionals
and related entropic values are used for deriving relativistic geometric
evolution flow equations. For self-similar configurations, such equations
describe generalized Ricci solitons defining modified Einstein equations. We
analyze possible connections between relativistic models of nonholonomic Ricci
flows and emergent modified gravity theories. We prove that corresponding
systems of nonlinear partial differential equations, PDEs, for entropic flows
and modified gravity possess certain general decoupling and integration
properties. There are constructed new classes of exact and parametric solutions
for nonstationary configurations and locally anisotropic cosmological metrics
in modified gravity theories and general relativity. Such solutions describe
scenarios of nonlinear geometric evolution and gravitational and matter field
dynamics with pattern-forming and quasiperiodic structure and various space
quasicrystal and deformed spacetime crystal models. We analyze new classes of
generic off-diagonal solutions for entropic gravity theories and show how such
solutions can be used for explaining structure formation in modern cosmology.
Finally, we speculate why the approaches with Perelman-Lyapunov type
functionals are more general or complementary to the constructions elaborated
using the concept of Bekenstein-Hawking entropy.
