Grain boundaries have a more open structure.
However, a FLUX is the mass transport through a unit area, in a unit time. If you take a random section through the microstructure, the area presented by the lattice is far greater than than presented by the grain boundaries. Therefore, at high temperatures where the difference between lattice diffusivity and boundary diffusivity is relatively small, the larger area presented by the lattice more than makes up for the smaller lattice diffusivity. Hence, the net mass transport consists more of diffusion through the lattice than through the grain boundaries. At low temperatures, the boundary diffusivity is many orders of magnitude greater than lattice diffusivity, so the greater cross-section presented by the lattice does not compensate for its low diffusivity.