I don’t think so, as being colored blue at room temperature and burning blue are different things. If you think about it, when you burn colored paper you don’t get a correspondingly colored flame.

The pigment is an organic molecule so it'd burn the same color as other organic molecules as the fire breaks it down https://en.m.wikipedia.org/wiki/Xylindein. Flame color is based on the elements in the flame. https://en.m.wikipedia.org/wiki/Flame_test