The temperature dependences of the electronic structure, spin-resolved photo-emission, and magnetic linear dichroism of the prototypical system 2 ML Co on Cu(001) (ML stands for monolayer(s)) are presented in a detailed theor-etical study. Relativistic ab initio electronic structure calculations were carried out within the framework of multiple-scattering theory, the spin-resolved photoemission was calculated within the one-step model. The temperature dependence was taken into account within the disordered local moment picture. The spectral features show a distinct dispersion and broadening with temperature, in particular those derived from quantum-well states in the Co film. These findings can be traced back to the layer- and spin-resolved Bloch spectral function. Further, Co-derived maxima in the photoemission intensities behave significantly differently with temperature to Cu-derived ones. The spin-resolved photoemission intensities compare well with experimental data and with theoretical data obtained within the fluctuating local moment picture. Magnetic linear dichroism in spin-resolved photoemission is discussed in terms of asymmetries which are related to the spin polarizations.