Context. Thermal annealing of interstellar ices takes place in several stages of star formation. Knowledge of this process comes from a combination of astronomical observations and laboratory simulations under astrophysically relevant conditions. Aims. For the first time we present the results of temperature programmed desorption (TPD) experiments with pre-cometary ice analogs composed of up to five molecular components: H 2 O, CO, CO 2 , CH 3 OH, and NH 3 . Methods. The experiments were performed with an ultra-high vacuum chamber. A gas line with a novel design allows the controlled preparation of mixtures with up to five molecular components. Volatiles desorbing to the gas phase were monitored using a quadrupole mass spectrometer, while changes in the ice structure and composition were studied by means of infrared spectroscopy. Results. The TPD curves of water ice containing CO, CO 2 , CH 3 OH, and NH 3 present desorption peaks at temperatures near those observed in pure ice experiments, volcano desorption peaks after water ice crystallization, and co-desorption peaks with water. Desorption peaks of CH 3 OH and NH 3 at temperatures similar to the pure ices takes place when their abundance relative to water is above ∼3% in the ice matrix. We found that CO, CO 2 , and NH 3 also present co-desorption peaks with CH 3 OH, which cannot be reproduced in experiments with binary water-rich ice mixtures. These are extensively used in the study of thermal desorption of interstellar ices. Conclusions. These results reproduce the heating of circumstellar ices in hot cores and can be also applied to the late thermal evolution of comets. In particular, TPD curves represent a benchmark for the analysis of the measurements that mass spectrometers on board the ESA-Rosetta cometary mission will perform on the coma of comet 67P/Churyumov-Gerasimenko, which will be active before the arrival of Rosetta according to our predictions.