Good point. However in most cases, if they have a 2” diagonal, the diagonal is going to take longer to cool than the doublet objective, making this somewhat irrelevant.
The difference in cooling between an ED doublet and an achromat of the same aperture, would be because typically the ED will be faster, requiring thicker lenses. More glass, longer cooling.
Certainly I wouldn’t decide which doublet refractor to get based on cooling. I might lean towards a doublet vs triplet. But the difference between two doublets is likely to be small, and overshadowed by the diagonal anyway.