 Averaging Kähler forms over a complex torus

Averaging a Kähler form over translations with normalized Haar measure gives an invariant Kähler form in the same de Rham cohomology class. Positivity is preserved by averaging positive values on $(v,Jv)$, and the class is unchanged because translations are homotopic to the identity. Thus any Hodge metric on a complex torus can be replaced by an invariant one.