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Using many-body perturbation theory and density functional perturbation theory, we study the vibrational spectra of the internal stretch (IS) mode of CO on Pd(111) for the bridge and hollow adsorption structures that are experimentally identified at 0.5~ML coverage. Our theoretical treatment allows us to determine the temperature dependence of the IS vibrational spectra under thermal conditions as well as the time evolution of the non-equilibrium transient spectra induced by femtosecond laser pulses. Under thermal conditions (i.e., for equal electronic $T_e$ and phononic $T_l$ temperatures), the calculated lifetimes at 10-150~K are mostly due to nonadiabatic couplings (NC), i.e., first-order electronic excitations. As temperature increases, also the contribution of the second-order electron mediated phonon-phonon couplings (EMPPC) progressively increases from 25\% at low temperatures to 50\% at 300~K. Our calculations for the laser-induced non-equilibrium conditions comprise experimental absorbed fluences of 6-130~J/m$^2$. For fluences for which $T_e>$2000~K, the transient vibrational spectra are characterized by two different regimes that follow the distinct time-evolution of $T_e$ and $T_l$ and are respectively dominated by NC and EMPPC processes. At lower fluences, the initial fast regime becomes progressively negligible as $T_e$ decreases and only the steady second regime remains visible. Qualitatively, all these spectral properties are common to the both adsorption structures studied here., 16 pages, 10 figures, 3 tables