Ciao Francesco!
In general if a projective manifold $X$ has ample canonical bundle, then it admits a Kähler-Einstein metric $\omega$ of negative Einstein constant, i.e. a Kähler metric such that $\operatorname{Ric}(\omega)=-\omega$ (by the Aubin-Yau Theorem).
In particular $T_X$ admits a Hermite-Einstein metric, and then so does any irreducible $\operatorname{GL}(T_X)$-representation, such as the symmetric powers of the cotangent bundle. This makes these vector bundles $[\omega]$-semistable, by the easy direction of the Kobayashi-Hitchin correspondence.
Finally, in this case, begin $[\omega]$-semistable means that these vector bundle are $K_X$-semistable, since $[\omega]=[-\operatorname{Ric}(\omega)]=c_1(K_X)$.
You can see all this and much more on S. Kobayashi "Differential geometry of complex vector bundles".