  The Euler characteristic and the volume are two best-known multiplicative
invariants of manifolds under finite covers. On the other hand, quantum
invariants of 3-manifolds are not multiplicative. We show that a perturbative
power series, introduced by Dimofte and the first author and shown to be a
topological invariant of cusped hyperbolic 3-manifolds by Storzer--Wheeler and
the first author, and conjectured to agree with the asymptotics of the Kashaev
invariant to all orders in perturbation theory, is asymptotically
multiplicative under cyclic covers. Moreover, its coefficients are determined
by polynomials constructed out of twisted Neumann--Zagier data. This gives a
new $t$-deformation of the perturbative quantum invariants, different than the
$x$-deformation obtained by deforming the geometric representation. We
illustrate our results with several hyperbolic knots.
