  Thermal stability of the Topological Color Code in presence of a thermal bath
is studied. We study the Lindblad evolution of the observables in the weak
coupling limit of the Born-Markov approximation. The auto-correlation functions
of the observables are used as a figure of merit for the thermal stability. We
show that all of the observables auto-correlation functions decay exponentially
in time. By finding a lower bound of the decay rate, which is a constant
independent of the system size, we show that the Topological Color Code is
unstable against thermal fluctuations from the bath at finite temperature, even
though it is stable at T=0 against local quantum perturbations.
