Spin pumping from ferromagnetic Fe into antiferromagnetic CoO across a Ag spacer layer was studied using ferromagnetic resonance (FMR) in Py/CoO/Ag/Fe/Ag(001). The thin Py film on top of CoO permits an alignment of the CoO Neél vector through field cooling in two otherwise equivalent  and  crystalline axes which are parallel and perpendicular to the Fe magnetization direction, respectively. Fe FMR linewidth is measured as a function of Ag thickness in 10-20-GHz frequency range and in 180-330 K temperature range. We find that there exists an anisotropy in the Fe FMR damping for parallel and perpendicular alignment of the Fe and CoO spins. However, such anisotropic damping exists only at thin Ag thickness where there exists a magnetic interlayer coupling between Fe and CoO, and vanishes at thick Ag thickness where the interlayer coupling becomes negligible but permitting spin-current transmission into CoO. Our result indicates the absence of anisotropic spin current for parallel and perpendicular alignment of the Fe and CoO spin axes.