We examine hilltop quintessence models , in which the scalar field is rolling near a local maximum in the potential , and w \approx - 1 .
We first derive a general equation for the evolution of \phi in the limit where w \approx - 1 .
We solve this equation for the case of hilltop quintessence to derive w as a function of the scale factor ; these solutions depend on the curvature of the potential near its maximum .
Our general result is in excellent agreement ( \delta w \lesssim 0.5 \% ) with all of the particular cases examined .
It works particularly well ( \delta w \lesssim 0.1 \% ) for the pseudo-Nambu-Goldstone Boson potential .
Our expression for w ( a ) reduces to the previously-derived slow-roll result of Sen and Scherrer in the limit where the curvature goes to zero .
Except for this limiting case , w ( a ) is poorly fit by linear evolution in a .