It's strictly related to the weight. If you can double the size and keep the weight down to only double instead of 8 times then you only need double the power to achieve the same climb. However since you're squaring the frontal area and wetted surface there will be a big increase in the power needed to maintain level flight. It'll be a blend of the power needed to reach the new flying speed as well as to combat the increase in section drag. If you were able to only let the weight double the flying speed would actually not change. Or rather it may well reduce a little due to the improvement in Reynolds number of the wider wing chord which would allow the slightly slower speed. So the only required increase would be in the frontal area and wetted surface drag.
More realistically your weight will fall somewhere between the max of 8 times higher suggested by the simiple scaling factor and the much harder to achieve mere doubling that would only come from doing a complete structural re-design. Where the final numbers would land up is totaly up in the air.... sorry, bad pun
Models climb based on watts per pound for electrics. This factor is pretty common over a fairly wide range of sizes from all I've seen. It pretty much stands to reason that to maintain level flight also requires a fairly consistent value of watts per pound assuming that it's a design that is more like a sailplane or other smaller fuselage and generous wing area style model.
I can see from this thread and the other one that you're keen on really trying to nail down this issue with pure numbers. While I'm sure it can be done there's no real overall formula that will do it down to the last picowatt. There's just too many variables to nail it down that tight. Frontal area and wetted surface gives us a good impression but if you achieve minimal frontal area and wetted surface with shapes that promote poor airflow and result in large turbulent wakes then you hurt the design more than if you use a simple but clean shape. And we just do not have any equations commonly available for describing or avoiding this sort of issue. Sailplanes get around the problem by making their fuselage as skinny as possible while staying clean. A cargo carrying model doesn't have that luxary. So we come back to basics and the idea that a shape that uses an airfoil for the side and top view is going to generate a smoother airflow without any gross separation bubbles at the expense of a little more wetted surface.