  Let $X$ be a complex toric variety equipped with the action of an algebraic
torus $T$, and let $G$ be a complex linear algebraic group. We classify all
$T$-equivariant principal $G$-bundles $\mathcal{E}$ over $X$ and the morphisms
between them. When $G$ is connected and reductive, we characterize the
equivariant automorphism group $\text{Aut}_T(\mathcal{E} )$ of $\mathcal{E}$ as
the intersection of certain parabolic subgroups of $G$ that arise naturally
from the $T$-action on $\mathcal{E}$. We then give a criterion for the
equivariant reduction of the structure group of $\mathcal{E}$ to a Levi
subgroup of $G$ in terms of $\text{Aut}_T(\mathcal{E} )$. We use it to prove a
principal bundle analogue of Kaneyama's theorem on equivariant splitting of
torus equivariant vector bundles of small rank over a projective space. When
$X$ is projective and $G$ is connected and reductive, we show that the notions
of stability and equivariant stability are equivalent for any $T$-equivariant
principal $G$-bundle over $X$.
