There are distinct differences in the alloyed golds with respect to machining and polishing. Rose gold and white gold do require more effort to achieve the standard of finish Patek requires. Of course, platinum is an entirely different animal altogether with even more effort required.
Maybe demand has some influence but I would suspect the additional cost is not related as much to demand as to cost of production.
My refiner produced a small batch of purple gold (aluminum is the alloy) for us to play with. Talk about an infuriating alloy! You cannot bend it, can barely drill it, and if dropped, can shatter. But as a blob that you can't do much with other than looking at it, great color when polished. Spent hours, destroyed tools, and took more Tylenol imaginable.
All of these alloyed metals present their own unique properties that have to be dealt with individually in order to produce the product, which to your eye is no different from each other, except the color.