Using density functional theory calculations, we investigate the surface deposition of O on Co(111) terrace and Co(221), Co(112) step surfaces in realistic Fischer Tropsch (FT) reaction conditions. We compared these results with previous works for C deposition on Co surfaces. Furthermore, we compute the activation barriers of C and O diffusion in subsurface Co(111) sites at different surface coverages. According to our results, thermodynamics and kinetics exhibit opposite trends, and the final cause of the catalyst deactivation is the outcome of a trade-off between the two effects. In particular, we show that the surface deposition of O is thermodynamically favorable. However, in order to increase a surface coverage beyond 1 ML, the O atom should first diffuse within the Co sublattice, which is kinetically hindered compared to subsurface C diffusion. However, the formation of an oxide shell around Co is possible from a thermodynamic and a kinetic point of view, particularly in the presence of an oxygen surface vacancy allowing the outward relaxation of Co atom (Kirkendall-type process). Lastly, we discuss in which circumstances the formation of this O overlayer prevails over the deactivation by the deposition of C resulting from a subtle interplay between the kinetics of the FT reaction steps.