Big History provides a unique opportunity to consider the development of the Universe as a single process. Within Big History studies one can distinguish some common evolutionary laws and principles. However, it is very important to recognize that there are many more such integrating principles, laws, mechanisms and patterns of evolution at all its levels than it is usually supposed. In the meantime, we can find the common traits in development, functioning, and interaction of apparently rather different processes and phenomena of Big History. Of special importance is the point that many principles, patterns, regularities, and rules of evolution, which we tend to find relevant only for the biological and social levels of evolution, may be also applied to the cosmic phase of evolution. The present article attempts (within such a framework for the first time in the Big History framework) at combining Big History potential with the potential of Evolutionary Studies. It does not only analyze the history of the Cosmos. It studies similarities between evolutionary laws, principles, and mechanisms at various levels and phases of Big History. Such an approach opens up some new perspectives for our understanding of evolution and Big History, their driving forces, vectors, and trends; it creates a consolidated field for interdisciplinary research.