Summary: Recent large scale simulations indicate that a powerful goodness-of-fit test for copulas can be obtained from the process comparing the empirical copula with a parametric estimate of the copula derived under the null hypothesis. A first way to compute approximate \(p\)-values for statistics derived from this process consists of using the parametric bootstrap procedure recently thoroughly revisited by C. Genest and B. Rémillard [Ann. Inst. Henri Poincaré, Probab. Stat. 44, No. 6, 1096–1127 (2008; Zbl 1206.62044)]. Because it heavily relies on random number generation and estimation, the resulting goodness-of-fit test has a very high computational cost that can be regarded as an obstacle to its application as the sample size increases. An alternative approach proposed by the authors consists of using a multiplier procedure. The study of the finite-sample performance of the multiplier version of the goodness-of-fit test for bivariate one-parameter copulas showed that it provides a valid alternative to the parametric bootstrap-based test while being orders of magnitude faster. The aim of this work is to extend the multiplier approach to multivariate multiparameter copulas and study the finite-sample performance of the resulting test. Particular emphasis is put on elliptical copulas such as the normal and the \(t\) as these are flexible models in a multivariate setting. The implementation of the procedure for the latter copulas proves challenging and requires the extension of the Plackett formula for the \(t\)-distribution to arbitrary dimension. Extensive Monte Carlo experiments, which could be carried out only because of the good computational properties of the multiplier approach, confirm in the multivariate multiparameter context the satisfactory behavior of the goodness-of-fit test.