Aims. Using high-quality, broad-band afterglow data for GRB 091127, we investigate the validity of the synchrotron fireball model for gamma-ray bursts (GRBs), and infer physical parameters of the ultra-relativistic outflow. Methods. We used multi-wavelength (NIR to X-ray) follow-up observations obtained with GROND simultaneously in the g r i z JH filters and the XRT onboard the Swift satellite in the 0.3 to 10 keV energy range. The resulting afterglow light curve is of excellent accuracy with relative photometric errors as low as 1%, and the spectral energy distribution (SED) is well-sampled over 5 decades in energy. These data present one of the most comprehensive observing campaigns for a single GRB afterglow and allow us to test several proposed emission models and outflow characteristics in unprecedented detail.Results. Both the multi-color light curve and the broad-band SED of the afterglow of GRB 091127 show evidence of a cooling break moving from high to lower energies. The early light curve is well described by a broken power-law, where the initial decay in the optical/NIR wavelength range is considerably flatter than at X-rays. Detailed fitting of the time-resolved SED shows that the break is very smooth with a sharpness index of 2.2 ± 0.2, and evolves towards lower frequencies as a power-law with index −1.23 ± 0.06. These are the first accurate and contemporaneous measurements of both the sharpness of the spectral break and its time evolution. Conclusions. The measured evolution of the cooling break (ν c ∝ t ∼−1.2 ) is not consistent with the predictions of the standard model, wherein ν c ∝ t ∼−0.5 is expected. A possible explanation for the observed behavior is a time dependence of the microphysical parameters, in particular the fraction of the total energy in the magnetic field B . This conclusion provides further evidence that the standard fireball model is too simplistic, and time-dependent micro-physical parameters may be required to model the growing number of well-sampled afterglow light curves.