  We elaborate on statistical thermodynamics models of relativistic geometric
flows as generalisations of G. Perelman and R. Hamilton theory centred around
C. Carath\'{e}odory axiomatic approach to thermodynamics with Pfaffian
differential equations. The anholonomic frame deformation method, AFDM, for
constructing generic off--diagonal and locally anisotropic cosmological
solitonic solutions in the theory of relativistic geometric flows and general
relativity is developed. We conclude that such solutions can not be described
in terms of the Hawking--Bekenstein thermodynamics for hypersurface,
holographic, (anti) de Sitter and similar configurations. The geometric
thermodynamic values are defined and computed for nonholonomic Ricci flows,
(modified) Einstein equations, and new classes of locally anisotropic
cosmological solutions encoding solitonic hierarchies.
