Black Holes with MDRs and Bekenstein-Hawking and Perelman Entropies for Finsler-Lagrange-Hamilton Spaces

New geometric and analytic methods for generating exact and parametric solutions in generalized Einstein-Finsler like gravity theories and nonholonomic Ricci soliton models are reviewed and developed. We show how generalizations of the Schwarzschild - (anti) de Sitter metric can be constructed for modified gravity theories with arbitrarily modified dispersion relations, MDRs, and Lorentz invariance violations, LIVs. Such theories can be geometrized on cotangent Lorentz bundles (phase spaces) as models of relativistic Finsler-Lagrange-Hamilton spaces. There are considered two general classes of solutions for gravitational stationary vacuum phase space configurations and nontrivial (effective) matter sources or cosmological constants. Such solutions describe nonholonomic deformations of conventional higher dimension black hole, BH, solutions with general dependence on effective four-dimensional, 4-d, momentum type variables. For the first class, we study physical properties of Tangherlini like BHs in phase spaces with generic dependence on an energy coordinate/ parameter. We investigate also BH configurations on base spacetime and in curved cofiber spaces when the BH mass and the maximal speed of light determine naturally a cofiber horizon. For the second class, the solutions are constructed with Killing symmetry on an energy type coordinate. There are analysed the conditions when generalizations of Beckenstein-Hawking entropy (for solutions with conventional horizons) and/or Grigory Perelman's W-entropy (for more general generic off-diagonal solutions) can be defined for phase space stationary configurations.