  The aim of this paper is to study the graded limits of minimal affinizations
over the quantum loop algebra of type $G_2$. We show that the graded limits are
isomorphic to multiple generalizations of Demazure modules, and obtain defining
relations of them. As an application, we obtain a polyhedral multiplicity
formula for the decomposition of minimal affinizations of type $G_2$ as a
$U_q(\mathfrak{g})$-module, by showing the corresponding formula for the graded
limits.
