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We consider cosmological dynamics in generalized modified gravity theory with the $R\Box R$ term added to the action of the form $R+R^N$. Influence of $R \Box R$ term to the known solutions of modified gravity is described. We show that in particular case of $N=3$ these two non-Einstein terms are equally important on power-law solutions. These solutions and their stability have been studied using dynamical system approach. Some results for the case of $N \ne 3$ (including stability of de Sitter solution in the theory under investigation) have been found using other methods.

Viable models of modified gravity which satisfy both local as well as cosmological tests are investigated. It is demonstrated that some versions of such highly non-linear models exhibit multiply de Sitter universe solutions, which often appear in pairs, being one of them stable and the other unstable. It is explicitly shown that, for some values of the parameters, it is possible to find several de Sitter spaces (as a rule, numerically); one of them may serve for the inflationary stage, while the other can be used for the description of the dark energy epoch. The numerical evolution of the effective equation of state parameter is also presented, showing that these models can be considered as natural candidates for the unification of early-time inflation with late-time acceleration through dS critical points. Moreover, based on the de Sitter solutions, multiply SdS universes are constructed which might also appear at the (pre-)inflationary stage. Their thermodynamics are studied and free energies are compared., Comment: 25 pages, 8 figures, 2 references added; PRD accepted version