  Let $f$ be an exact area-preserving monotone twist diffeomorphism of the
infinite cylinder and $P_{\omega,f}(\xi)$ be the associated Peierls barrier. In
this paper, we give the H\"{o}lder regularity of $P_{\omega,f}(\xi)$ with
respect to the parameter $f$. In fact, we prove that if the rotation symbol
$\omega\in
(\mathbb{R}\setminus\mathbb{Q})\bigcup(\mathbb{Q}+)\bigcup(\mathbb{Q}-)$, then
$P_{\omega,f}(\xi)$ is $1/3$-H\"{o}lder continuous in $f$, i.e.
$$|P_{\omega,f'}(\xi)-P_{\omega,f}(\xi)|\leq C\|f'-f\|_{C^1}^{1/3} ,~~\forall
\xi\in\mathbb{R}$$ where $C$ is a constant. Similar results also hold for the
Lagrangians with one and a half degrees of freedom. As application, we give an
open and dense result about the breakup of invariant circles.
