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New geometric and analytic methods for generating exact and parametricsolutions in generalized Einstein-Finsler like gravity theories andnonholonomic Ricci soliton models are reviewed and developed. We show howgeneralizations of the Schwarzschild - (anti) de Sitter metric can beconstructed for modified gravity theories with arbitrarily modified dispersionrelations, MDRs, and Lorentz invariance violations, LIVs. Such theories can begeometrized on cotangent Lorentz bundles (phase spaces) as models ofrelativistic Finsler-Lagrange-Hamilton spaces. There are considered two generalclasses of solutions for gravitational stationary vacuum phase spaceconfigurations and nontrivial (effective) matter sources or cosmologicalconstants. Such solutions describe nonholonomic deformations of conventionalhigher dimension black hole, BH, solutions with general dependence on effectivefour-dimensional, 4-d, momentum type variables. For the first class, we studyphysical properties of Tangherlini like BHs in phase spaces with genericdependence on an energy coordinate/ parameter. We investigate also BHconfigurations on base spacetime and in curved cofiber spaces when the BH massand the maximal speed of light determine naturally a cofiber horizon. For thesecond class, the solutions are constructed with Killing symmetry on an energytype coordinate. There are analysed the conditions when generalizations ofBeckenstein-Hawking entropy (for solutions with conventional horizons) and/orGrigory Perelman's W-entropy (for more general generic off-diagonal solutions)can be defined for phase space stationary configurations.

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