Based on the coherence theory of non-stationary fields, the closed-form expression for the cross-spectral density matrix of spatially and spectrally partially coherent electromagnetic cosh-Gaussian pulsed beams propagating in free space is derived. It is shown that the spectral density, the spectral degree of polarization and the spectral degree of coherence of electromagnetic cosh-Gaussian pulsed beams depend on the position of the field point and the beam parameters, such as the decentered parameter, spatial correlation length and temporal coherence length. Spatially and spectrally partially coherent electromagnetic Gaussian-Schell-model pulsed beams, spectrally fully coherent electromagnetic pulsed Gaussian beams and spatially fully coherent electromagnetic Gaussian pulsed beams can be treated as special cases of our results.