Let $R$ be an integral domain and let $f:R\to R$ be an automorphism of $R$. Is it always true that $\mathrm{Frac}(R^f) = [\mathrm{Frac}(R)]^f$ where $\mathrm{Frac}$ denotes the fraction field, and $(-)^f$ denotes the ring of fixed elements under $f$?

This is one of those things that sounds too good to be true. In fact, it is not always true. Take $\Q[x,y]$ and $f$ be the $\Q$-algebra automorphism given on generators by $x\mapsto 2x$ and $y\mapsto 2y$. Then $\Q[x,y]^f = \Q$ so the corresponding fraction field is $\Q$. Whereas, $(\mathrm{Frac} \Q[x,y])^f = \Q(x,y)^f$ contains the element $x/y$.