Here some quick sources:
From Mikko Folkersma’s PhD thesis
Figure 2.5 shows a comparison of the mass per projected area and the scaling of the commercial surf kites. Note that in kite surfing, the designed traction force of the kite remains approximately constant as the kite surface area grows. The larger kite designs are for lower wind velocities, and therefore the tensile forces in the membrane remain rather constant. While in AWE, scaling up aims to generate higher traction force, and therefore, the tensile forces increase linearly with the surface area. Then, the membrane thickness must increase, or more reinforcements and bridle lines are required. Hence, a similar downward trend for the specific mass can not be expected for the AWE kites when scaling up. Generally, a low specific mass is a desirable property as it allows operating the system at low wind speeds. However, lightweight wings are also more prone to disturbances such as rain and gusts, as mentioned earlier.
The mass of the TU Delft LEI V3 kite (25 m2 flat wing surface area, 19.7 m2 projected area) is 22.8 kg. This amounts to a mass/surface ratio of 1.16 kg/m2. Note that this includes bridle line system and kite control unit, which is 50% of the kite’s mass. So, for only the wing, we get to 0.58 kg/m2 which is in the range of conventional surf kites.
From Paulig et al (2013) we know
The specific weight of 160 to 320 m2 kites is around 0.5–0.6 kg/m2
which is in line with the V3 kite (I am not sure whether Skysails refers to flat or projected area). But this at least indicates that in this size range, the mass of the wing scales linearly with the wing surface area.
I do acknowledge that this is only a very rough analysis and I know that kites are increasingly reinforced and/or bridle lines added to increase the wing loading. So, a slightly steeper increase of mass than linear should be expected. But still, this is far from a square-cube law.