Equilibrium states for interval maps: the potential $-t\log |Df|$

Let $f:I \to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential $\phi_t:x\mapsto -t\log|Df(x)|$ for $t$ close to 1, and also that the pressure function $t \mapsto P(\phi_t)$ is analytic on an appropriate interval near $t = 1$.