  In this paper, we study the representations and module-extensions of hom
3-Lie algebras. We show that a linear map between hom 3-Lie algebras is a
morphism if and only if its graph is a hom 3-Lie subalgebra and show that the
derivations of a hom 3-Lie algebra is a Lie algebra. Derivation extension of
hom 3-Lie algebras are also studied as an application. Moreover, we introduce
the definition of $T_{\theta}$-extensions and $T^{*}_{\theta}$-extensions of
hom 3-Lie sub-algebras in terms of modules, provide the necessary and
sufficient conditions for $2k$-dimensional metric hom 3-Lie algebra to be
isomorphic to a $T^{*}_{\theta}$-extensions.
