Hi! Foremost, you have to understand that the colours that you see is acutally not what you think. For instance, you see red ice cube means that the ice cubes reflects only red colour to your eyes, and absorb all the other colours! Similarly, for black object, it's black cos all the colours are being absorbed, and no light is reflected to your eyes! So technically black is not even a colour!! (if you wear a black shirt in a hot day, u will feel extremely hot! That is cos your black shirt is absorbing all the light, and it's energy!)
Now, red is at the long wavelength of the visible specturm, meaning that it has the least energy. So if only red is reflected, all other high energy wavelength is absorbed, and these energy can therefore melt the ice. Similarly, blue has the shortest wavelength, but strongest energy in the specturm, but is reflected in a blue ice cube. So, blue ice cube will absorb other colours like red etc, but will not be as much as that of a red ice cube. So, blue will be slower to melth compared to the red.
In fact, I think it should melt according to the spectrum from red, orange, yellow, green, indigo, blue, violet, follow by regular ice cube. Regular ice cube is white, so it reflects all the visible light to your eyes, and does not absorb any light and hence will not absorb the energies that comes with the light.
Of cos, the above is based on the assumption that ur colouring are of the same material.. cos impurities can change the melting point of ice. Hope this helps!
Yes, colours with short wavelength has more energy (e.g. blue has more energy than red). But in blue ice cube, blue is reflected. Analogously, high energy light (e.g. blue) is reflected away, leaving low energy light to be absorbed by the blue ice cube. Red ice cube however, reflect away the low energy red light, but absorb the higher energy blue light. So, blue will melt slower than red.
Knowledge on physics of light