The theory of anharmonic coupling of two vibrations with vibrational quanta of the order of temperature of the experiment was developed. It is shown that even weak coupling can cause the pronounced temperature dependence of the corresponding infrared and resonance Raman bands. The theory is applied to the iron-imidazole resonance Raman band of the deoxyheme proteins and model compounds. It was shown that weak anharmonic coupling of the iron-imidazole and iron out-of-the-porphyrin-plane vibrations must lead to the movement of the peak position of the band to the red, its broadening and asymmetry upon increase of temperature. The stronger anharmonic coupling can even cause appearance of the structured iron-imidazole band at room temperature.