Rényi entropy is a generalization of the standard Shannon entropy and plays an important role in understanding complex dynamics of non-equilibrium systems. In line with [7], this brief report discusses the definition and application of Rényi entropy to Quantum Field Theory (QFT) and gravitational regime of primordial cosmology. In particular, we review the connection between Rényi entropy and entanglement entropy, on the one hand, and the maximal entropy principle (MEP), on the other. Remarkably, the latter enables derivation of the Higgs and W-boson masses in close agreement with experimental data. Appealing to multifractal analysis and the concept of generalized dimensions, we further bridge the divide between Rényi entropy and the "sum-of-squares" relationship of particle physics, corresponding to maximally entropy at index q=0. We emphasize the deep analogy between QFT entanglement and long-range correlations of complex dynamics and indicate that the MEP aligns with the universal route to chaos described by the Feigenbaum scenario. Finally, we review how Rényi entropy can be used to derive the four dimensionality of classical spacetime for q=1/2.