Applying the anholonomic frame deformation method, we construct various classes of cosmological solutions for effective Einstein -- Yang-Mills -- Higgs, and two measure theories. The types of models considered are Freedman-Lema\^{i}tre-Robertson-Walker, Bianchi, Kasner and models with attractor configurations. The various regimes pertaining to plateau--type inflation, quadratic inflation, Starobinsky type and Higgs type inflation are presented.