Arguments towards a quadratic relationship : A powerline designed with twice the voltage has everything larger, higher pylons, thicker conductors.
Making the conductors thicker is not necessary to design for higher voltage. (Making the insulators thicker, is required)
If you make the conductors thicker then you are changing the design to allow for higher current.
If you design for both higher voltage and higher current, then yes, the power carrying capability will increase more than if you just design for higher voltage.
As the designer you have the freedom to do this in any proportion, you don't have to double the current if you double the voltage. You could even design for lower current at higher voltage, keeping the power the same (for example, this might save you on the cost of the wires).
If you're talking about cross-country transmission lines you might be constrained by things like the weight-bearing capacity of the pylons, wind loading, etc.
Arguments towards a cubic relationship : If the wires of a power line for twice the voltage have their diameter twice larger, it means their surface is 4 times as much, thus supporting 4 times the current, hence 8 times the power.
It's not the total surface area, but the surface area per unit length, (aka, the circumference) that matters. And the circumference only increases in proportion to the diameter, not as the square of the diameter.