Any color other than white will get hotter in the sun. However with thick and good quality insulation, the difference in energy that actually transfers through the insulation into the cooler might be insignificant.
However, and this may actually be where the performance evens out, is that a color other than white will have a higher emissivity, meaning it radiates energy (cools down) faster once it stops absorbing energy (from the sun). So as soon as you take your other-than-white colored cooler out of the sun, it will cool off faster. And the insulation behind the exterior will cool off faster because the outside will be radiating more energy.
I bet the difference in performance over several days in 8 hours of direct sunlight would be hours worth of ice. And the number of times you open and close your lid, and the way you pack your stuff, will probably be a much bigger factor.
It would be cool to do a heat transfer simulation of 8 hours of sunlight with 8 hours of no sunlight and compare different colored surfaces with identical insulation backing and look at a graph of heat transfer over time, with a total amount of energy absorbed at the end. Anyone have time and means to do that?