I'm pretty sure power scales quadratically not exponentially with voltage, right?
That said, at the manufacturing scales of these parts leakage in the transistor can be as big a power consumer as the switching.
Leakage current increasing as voltage increases at a rate that is of the order ~e^Vth
I wouldn't really say its strictly either exponential or quadratic, you'd have to know a lot more about the process and the implementation to know how significant leakage current is in the design.
The static power dissipation from leakage currents though is Ileak = Isat * (e(qV/kT)-1).
http://www.siliconintelligence.com/people/binu/perception/no...