I develop the theory which should describe our universe evolution as a whole. My main purpose is  to resolve the fundamental puzzle of modern cosmology: why and how the universe accelerates? What is Dark Energy and Inflation? In fact, I proposed the first modified gravity which may describe the consistent universe history. In relation with modified gravity we proposed new very general and non-singular   entropy which generalises known earlier variants.  The alternative gravity which we develop should describe the whole sequence of the universe eras: inflation, radiation-matter dominance and dark energy era in a unified way. We also continue the search for exotic intermediate eras like pre-inflationary or exotic reheating era and the related gravitational waves. Our study of current universe  will clarify its future: if it will expand eternally or if its evolution will be finished in the future singularity.