From one year (2004) of thermosphere total density data inferred from CHAMP/STAR accelerometer measurements, we calculate the global thermosphere response to auroral magnetic activity forcing at middle and low latitudes using a method based on a singular value decomposition of the satellite data. This method allows separating the large‐scale spatial variations in the density, mostly related to altitude/latitude variations and captured by the first singular component, from the time variations, down to timescales on the order of the orbital period, which are captured by the associated projection coefficient. This projection coefficient is used to define a disturbance coefficient that characterizes the global thermospheric density response to auroral forcing. For quiet to moderate magnetic activity levels (Kp < 6), we show that the disturbance coefficient is better correlated with the magnetic am indices than with the magnetic ap indices. The latter index is used in all empirical thermosphere models to quantify the auroral forcing. It is found that the NRLMSISE‐00 model correctly estimates the main features of the thermosphere density response to geomagnetic activity, i.e., the morphology of Universal Time variations and the larger relative increase during nighttime than during daytime. However, it statistically underestimates the amplitude of the thermosphere density response by about 50%. This underestimation reaches 200% for specific disturbed periods. It is also found that the difference between daytime and nighttime responses to auroral forcing can statistically be explained by local differences in magnetic activity as described by the longitude sector magnetic indices.