  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.
