  In this paper, we investigate the Hamiltonian-stability of Lagrangian tori in
the complex hyperbolic space $\mathbb{C}H^n$. We consider a standard
Hamiltonian $T^n$-action on $\mathbb{C}H^n$, and show that every Lagrangian
$T^n$-orbits in $\mathbb{C}H^n$ is H-stable when $n\leq 2$ and there exist
infinitely many H-unstable $T^n$-orbits when $n\geq 3$. On the other hand, we
prove a monotone $T^n$-orbit in $\mathbb{C}H^n$ is H-stable and rigid for any
$n$. Moreover, we see almost all Lagrangian $T^n$-orbits in $\mathbb{C}H^n$ are
not Hamiltonian volume minimizing when $n\geq 3$ as well as the case of
$\mathbb{C}^n$ and $\mathbb{C}P^n$.
