Even among the most irradiated gas giants, so-called ultra-hot Jupiters, KELT-9b stands out as the hottest planet thus far discovered with a dayside temperature of over 4500K. At these extreme irradiation levels we expect an increase in heat redistribution efficiency and a low Bond albedo owed to an extended atmosphere with molecular hydrogen dissociation occurring on the planetary dayside. We present photometric observations of the KELT-9 system throughout 4 full orbits and 9 separate occultations obtained by the 30cm space telescope CHEOPS. The CHEOPS bandpass, located at optical wavelengths, captures the peak of the thermal emission spectrum of KELT-9b. In this work we simultaneously analyse CHEOPS phase curves as well as public phase curves from TESS and Spitzer to infer joint constraints on the phase curve variation, gravity-darkened transits and occultation depth in three bandpasses. We derive 2D temperature maps of the atmosphere at three different depths, finding that temperature increases at higher probed altitudes, i.e. a thermal inversion. The day-night heat redistribution of ~0.3 confirms expectations of enhanced energy transfer to the planetary nightside due to dissociation and recombination of molecular hydrogen. We also find a Bond albedo consistent with zero. We find no evidence of variability of the brightness temperature of the planet, excluding variability greater than 1% (1 sigma).