  We study geometric relativistic flow and Ricci soliton equations which (for
respective nonholonomic constraints and self-similarity conditions) are
equivalent to the gravitational field equations of $R^2$ gravity and/or to the
Einstein equations with scalar field in general relativity, GR. Perelman's
functionals are generalized for modified gravity theories, MGTs, which allows
to formulate an analogous statistical thermodynamics for geometric flows and
Ricci solitons. There are constructed and analyzed generic off-diagonal black
ellipsoid, black hole and solitonic exact solutions in MGTs and GR encoding
geometric flow evolution scenarios and nonlinear parametric interactions. Such
new classes of solutions in MGTs can be with polarized and/or running
constants, nonholonomically deformed horizons and/or imbedded self-consistently
into solitonic backgrounds. They exist also in GR as generic off-diagonal
vacuum configurations with effective cosmological constant and/or mimicking
effective scalar field interactions. Finally, we compute Perelman's energy and
entropy for black ellipsoids and evolution solitons in $R^2$ gravity.
