In an effort to derive temperature-based criteria of habitability for multicellular life, we investigated the thermal limits of terrestrial poikilotherms, i.e. organisms whose body temperature and the functioning of all vital processes is directly affected by the ambient temperature. Multicellular poikilotherms are the most common and evolutionarily ancient form of complex life on earth. The thermal limits for the active metabolism and reproduction of multicellular poikilotherms on earth are approximately bracketed by the temperature interval 0°C ≤ T ≤ 50°C. The same interval applies to the photosynthetic production of oxygen, an essential ingredient of complex life, and for the generation of atmospheric biosignatures observable in exoplanets. Analysis of the main mechanisms responsible for the thermal thresholds of terrestrial life suggests that the same mechanisms would apply to other forms of chemical life. We therefore propose a habitability index for complex life, h
050, representing the mean orbital fraction of planetary surface that satisfies the temperature limits 0°C ≤ T ≤ 50°C. With the aid of a climate model tailored for the calculation of the surface temperature of Earth-like planets, we calculated h
050 as a function of planet insolation, S, and atmospheric columnar mass, N
atm, for a few earth-like atmospheric compositions with trace levels of CO2. By displaying h
050 as a function of S and N
atm, we built up an atmospheric mass habitable zone (AMHZ) for complex life. At variance with the classic habitable zone, the inner edge of the complex life habitable zone is not affected by the uncertainties inherent to the calculation of the runaway greenhouse limit. The complex life habitable zone is significantly narrower than the habitable zone of dry planets. Our calculations illustrate how changes in ambient conditions dependent on S and N
atm, such as temperature excursions and surface dose of secondary particles of cosmic rays, may influence the type of life potentially present at different epochs of planetary evolution inside the AMHZ.