Let \( W \) be a \( G \)-graded algebra over a field of characteristic zero, where \( G \) is a finite group. We develop a theory of generalized \( G \)-graded polynomial identities satisfied by any finite-dimensional \( W \)-algebra \( A \), by mean of the graded multiplier algebra of \( A \). In particular, we first prove that the graded generalized exponent exists and equals the ordinary one. Then, we explicitly compute the \( G \)-graded generalized identities of \( UT_2 \), the 2 × 2 upper triangular matrix algebra equipped with its canonical \( \mathbb Z_2 \)-grading, under all the possible graded \( W \)-actions. Finally, we exhibit examples of varieties of graded \( W \)-algebras with almost polynomial growth of the codimensions.