  We consider families of multimodal interval maps with polynomial growth of
the derivative along the critical orbits. For these maps Bruin and Todd have
shown the existence and uniqueness of equilibrium states for the potential
$\phi_t:x\mapsto-t\log|Df(x)|$, for $t$ close to 1. We show that these
equilibrium states vary continuously in the weak$^*$ topology within such
families. Moreover, in the case $t=1$, when the equilibrium states are
absolutely continuous with respect to Lebesgue, we show that the densities vary
continuously within these families.
