Thank you for your comment. I understand that the word arbitrary was not proper in that question.
I agree with you regarding Feynmann integrals. In my case it should be arbitrary integer $n\geqslant 3$.
Of course I can define it for some particular value of n, for instance n=3,4,5,6,7 and so on, but my question is whether is it possible to define a metric in a way where the particular n for dimension is not given initially. The structure of the metric is rather simple, this is a diagonal metric with spherical, flat or hyperbolic symmetry, as it usually takes place for various cosmological models. Actually, the problem is related to cosmological/black holes' models in Horndeski gravity in n-dimensional space (where n is of course positive integer >2, with special emphasis on n=4, but higher (and lower) dimensions are also examined).
Thank you once again.