  An explicit formula to compute the multiplicative anomaly or defect of
$\zeta$-regularized products of linear factors is derived, by using a Feynman
parametrization, generalizing Shintani-Mizuno formulas. Firstly, this is
applied on $n$-spheres, reproducing known results in the literature. Then, this
framework is applied to a closed Einstein universe at finite temperature,
namely $S^1_{\beta}\times S^{n-1}$. In doing so, it is shown that the standard
Casimir energy for GJMS operators coincides with the accumulated multiplicative
anomaly for the shifted Laplacians that build them up. This equivalence between
Casimir energy and multiplicative anomaly, unnoticed so far to our knowledge,
brings about a new turn regarding the physical significance of the
multiplicative anomaly, putting both now on equal footing. An emergent improved
Casimir energy, that takes into account the multiplicative anomaly among the
building Laplacians, is also discussed.
