The authors study numerically the spreading of damage in the Euclidean travelling salesman problem. There is a critical temperature T_c approximately=0.20-0.25 below which damage does not spread. They search for multifractal behaviour in the moments of the damage probability distribution. Multifractality is found in the frozen phase when the moments are monitored as a function of time. If they are evaluated with the number of cities, multifractal behaviour occurs in the chaotic phase.