The micromagnetic exchange stiffness is a critical parameter in numerical modeling of magnetization dynamics and reversal processes, yet the current literature reports a wide range of values even for such simple and widely used material as cobalt. Using the ab initio estimated Heisenberg parameters we calculate the low temperature micromagnetic exchange stiffness for hexagonal-close-packed (hcp) and face-centered-cubic cobalt. For hcp Co they are slightly different in the directions parallel and perpendicular to the c axis. We establish the exchange stiffness scaling relation with magnetization A(m)∼m1.8 valid for a wide range of temperatures. For hcp Co we find an anisotropic domain wall width in the range 24-29 nm which increases with temperature. The results form a critical input for large-scale temperature-dependent micromagnetics simulations and demonstrate the importance of correct parametrization for accurate simulation of magnetization dynamics.