let $(M,\omega)$ be a Kahler-Einstein toric manifold of complex dimension $m$. By toric manifold i mean a manifold that has an open dense subset $X$ biholomorphic to an algebraic torus $\mathbb{T}^{m}:=( \mathbb{C}^{*})^{m}$ and there is a holomorphic action \begin{equation} \alpha:\mathbb{T}^{m}\times M\rightarrow M \end{equation} that on $X$ restricts to the standard action of $\mathbb{T}^{m}$ on itself. My question is the following: is the Kahler Einstein metric $\omega$ automatically invariant under the action of $\mathbb{T}^{m}$? If it is a known result can someone tell me a reference?

Thank you in advance.