  We study new classes of generic off-diagonal and diagonal cosmological
solutions for effective Einstein equations in modified gravity theories, MGTs,
with modified dispersion relations, MDRs, encoding possible violations of
(local) Lorentz invariance, LIVs. Such MGTs are constructed for Lagrange
densities with two non-Riemannian volume forms (two-measure theories, TMTs) and
associated bimetric/ biconnection geometric structures. For conventional 2+2
splitting, we can describe such models in Finsler like variables, which is
important for elaborating geometric methods of constructing exact and
parametric solutions. Such formulations of general relativity, GR, and MGTs are
considered for Lorentz manifolds and their (co) tangent bundles, in brief,
FTMT. Off-diagonal metrics solving gravitational field equations in FTMTs are
determined by generating functions, effective sources and integration constants
and characterized by nonholonomic frame torsion effects. Restricting the class
of integration functions, we can extract torsionless and diagonal
configurations and model emergent cosmological theories with square scalar
curvature, $R^2$, when the global Weyl-scale symmetry is broken via nonlinear
dynamical interactions with nonholonomic constraints. In the physical
Einstein-Finsler frame, the constructions involve (i) nonlinear
re-parametrization symmetries of the generating functions and effective
sources; (ii) effective potentials for the scalar field with possible two flat
regions which allows a unified description of locally anisotropic and/or
isotropic early universe inflation related to acceleration cosmology and dark
energy; (iii) there are "emergent universes" described by (off-) diagonal
solutions for certain nonholonomic phases and parametric cosmological evolution
resulting in various inflationary phases; (iv) we can reproduce in two-measure
theories massive gravity effects.
