  We study the additive eigenvalues on changing domains, along with the
associated vanishing discount problems. We consider the convergence of the
vanishing discount problem on changing domains for a general scaling type
$\Omega_\lambda = (1+r(\lambda))\Omega$ with a continuous function $r$ and a
positive constant $\lambda$. We characterize all solutions to the ergodic
problem on $\Omega$ in terms of $r$. In addition, we demonstrate that the
additive eigenvalue $\lambda\mapsto c_{\Omega_\lambda}$ on a rescaled domain
$\Omega_\lambda = (1+\lambda)\Omega$ possesses one-sided derivatives
everywhere. Additionally, the limiting solution can be parameterized by a real
function, and we establish a connection between the regularity of this real
function and the regularity of $\lambda \mapsto c_{\Omega_\lambda}$. We provide
examples where higher regularity is achieved.
