The energy is proportional to both the height and the mass. The mass is proportional to the height itself. Hence the energy is proportional to the square of the height.

If the fill level is h, so the weighted average of the mass is at h/2,then

E = m * g * h / 2

m = V * r

V = A * h

E = A * r * g * h^2 / 2

m2 * kg/m3 * m/s2 * m2 = kg m2 / s2 = J

Yeah, I totally forgot that there is a height component in the mass as well.