In this talk I want to discuss results with Mike Todd on the existence and uniqueness of equilibrium states for multimodal maps f:I→I with polynomial growth rates of derivatives along the critical orbits. This requires more involved methods than for the Collet-Eckmann class where these derivatives grow exponential. The class of potentials include -t log|f'|, t ≈ 1, for which we prove the analyticity of the pressure function except for the phase transition occurring at t=1 for non-Collett-Eckmann maps.