The scaling solution of the functional flow in dilaton quantum gravity leads to highly predictive cosmology. Inflation and quintessence can both be described by a single scalar field. The cosmic time evolution of this cosmon field realizes a crossover from the region of an ultraviolet fixed point in the infinite past to an infrared fixed point in the infinite future. This amounts to a transition from early inflation to late dynamical dark energy, with intermediate radiation and matter domination. The scaling solution of the renormalization flow in quantum gravity connects the two fixed points. It provides for the essential characteristics of the scalar potential needed for the crossover cosmology and solves the cosmological constant problem dynamically. The quantum scale symmetry at the infrared fixed point protects the tiny mass of the cosmon and suppresses the cosmon coupling to atoms without the need of a non-linear screening mechanism, thereby explaining apparent issues of fine tuning. For a given content of particles the scaling solution of quantum gravity is a predictive framework for the properties of inflation and dynamical dark energy. No flow away from the scaling solution may be needed.