We consider the Curie-Weiss model at initial temperature 0
For initial temperature beta (-1)> 1 we prove that the time-evolved measure stays Gibbs forever, for any (possibly low) temperature of the dynamics.
In the regime of heating to low-temperatures from even lower temperatures, 0
In the regime of further cooling from low-temperatures, beta'(-1)
To our knowledge this is the first example of the rigorous study of non-Gibbsian phenomena related to cooling, albeit in a mean-field setup.