I'm sure the answer to the following question is well known but I couldn't find the answer I needed.

Let $(\Sigma,\sigma)$ denote the full shift on $k$ symbols and let $\mu$ be an invariant measure for which the system is ergodic. Let $f:\Sigma\to\mathbb R^+$ be a Holder continuous roof function. Let $$\Sigma_f:=\{(\underline a,t):\underline a\in\Sigma, 0\leq t\leq f(\underline a)\} $$ and let $\mu_f$ denote the measure $(\mu\times Leb)|_{\Sigma_f}$. Let $\phi$ denote the usual suspension flow on $\Sigma_F$, it is known that $(\Sigma_f, \phi,\mu_f)$ is ergodic.

Let $T:\Sigma_f\to\Sigma_f$ denote the time one map associated to $\phi$, $T$ preserves measure $\mu_f$. Under what conditions on $f$ is the system $(\Sigma_f, T, \mu_f)$ ergodic?

My guess is that the system is ergodic unless $f$ is cohomologous to a rational constant, but if anyone knows the correct statement and a reference for it that would be great.