Electron cloud (e-cloud) mitigation is an essential requirement for proton circular accelerators in order to guarantee beam stability at a high intensity and limit the heat load on cryogenic sections. Laser-engineered surface structuring is considered a credible process to reduce the secondary electron yield (SEY) of the surfaces facing the beam, thus suppressing the e-cloud phenomenon within the high luminosity upgrade of the LHC collider at CERN (HL-LHC). In this study, the SEY of Cu samples with different oxidation states, obtained either through laser treatment in air or in different gas atmospheres or via thermal annealing, has been measured at room and cryogenic temperatures and correlated with the surface composition measured by x-ray photoelectron spectroscopy. It was observed that samples treated in nitrogen display the lowest and more stable SEY values, correlated with the lower surface oxidation. In addition, the surface oxide layer of air-treated samples charges upon electron exposure at a low temperature, leading to fluctuations in the SEY.