Off-Diagonal Decoupling and Integrability of (Non) Metric Geometric Flow and Finsler-Lagrange-Hamilton Modified Einstein Equations

Many Finsler-like geometric and modified gravity theories (MGTs) have been elaborated during the last 70 years. They were defined by different types of Finsler generating functions and metric, or nonmetric, nonlinear and linear connections; and by postulating different types of fundamental geometric objects and related nonholonomic geometric evolution or dynamical equations. Certain proposed variants of locally anisotropic gravitational and matter field equations were not completely defined geometrically or, in other cases, elaborated for some particular models. We provide a status report, with historical remarks and a summary of new results and methods on Finsler-Lagrange-Hamilton (FLH) geometric flow and gravity theories, which can be constructed in general axiomatic form on (co) tangent Lorentz bundles (as modifications of Einstein gravity). Such models are characterized by nonlinear dispersion relations and may encode nonassociative and nocommutative corrections from string theory, quantum corrections, or contributions from various types of MGTs. To generate physically important solutions of the FLH modified Einstein equations, we formulated the anholonomic frame and connection deformation method, AFCDM. We provide a proof of the general integrability of such FLH geometric flow and MGTs and analyse new classes of physically important generic off-diagonal solutions. Such solutions are determined by respective classes of generating functions and generating sources depending, in principle, on all spacetime and (co) fiber coordinates. We discuss the physical properties of certain important examples of solutions for Finsler black holes, wormholes and locally anisotropic cosmological solutions constructed by applying the AFCDM. In general, such generic off-diagonal solutions/ scenarios do not involve certain hypersurface or holographic configurations and can't be described in the framework of the Bekenstein-Hawking thermodynamic paradigm. We argue that generalizing the concept of G. Perelman's entropy for relativistic FLH geometric flows allows us to define and compute new types of geometric thermodynamic variables characterizing different FLH theories and various classes of solutions.