Rational Homotopy and Hodge Theory of Moduli Stacks of principal $G$-bundles

For a semisimple complex algebraic group $G$ we determine the rational cohomology and the Hodge-Tate structure of the moduli stack $\mathcal{B}u{n}_{G,X}$ of principal $G$-bundles over a connected smooth complex projective variety $X$ of special type using the homotopy theory of the underlying topological stack.