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-beta(1) x Sn-1. In doing so, it is shown that the standard Casimir energy (as computed via zeta regularization) 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 within zeta regularization, 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 incorporates the multiplicative anomaly among the building Laplacians, is also discussed.