Herein, the authors introduce a class of multidimensional weighted Kantorovich operators \(K_n\), \(n\geq 1\), whose definition is given on the space of continuous functions \(C(Q_{d})\) (where \(Q_d\) is the \(d\)-dimensional hypercube \([0,1]^{d}\), \(d\geq 1\)), and it involves the well-known Bernstein polynomials. In this setting, they prove the existence of a unique probability measure on \(Q_d\) which is invariant with respect to \(K_n\), and they determine such a measure. Furthermore, the authors give a convergence result of the iterates \(K_n f\) to \(f\), uniformly on \(C(Q_d)\). Finally, they point out that the above class \(K_n\) is a generalization of some Kantotovich type operators, consequently their approach is unifying for the study of approximation results for \(K_n\).