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An $S$-unimodal map $f$ with flat critical point satisfying the Misiurewiczcondition displays a freezing phase transition in positive spectrum. We analyzestatistical properties of the equilibrium state $\mu_t$ for the potential$-t\log|Df|$, as well as how the phase transition slows down the rate of decayof correlations. We show that $\mu_t$ has exponential decay of correlations forall inverse temperature $t$ contained in the positive entropy phase$(t^-,t^+)$. If the critical point is not too flat, then the freezing point$t^+$ is equal to $1$, and the absolutely continuous invariant probabilitymeasure (acip for short) is the unique equilibrium state at the transition. Weexhibit a case in which the acip has sub-exponential decay of correlations and$\mu_t$ converges weakly to the acip as $t\nearrow t^+$.

Equilibrium states at freezing phase transition in unimodal maps with flat critical point