We study a class of multivariate tempered stable distributions and introduce the associated class of tempered stable Sato subordinators. These Sato subordinators are used to build additive inhomogeneous processes by subordination of a multiparameter Brownian motion. The resulting process is additive and time inhomogeneous. Furthermore, these processes are associated with the distribution at unit time of a class of Lévy process with good fit properties on fifinancial data. The main feature of the Sato subordinated Brownian motion is that it has time dependent correlation, whereas the Lévy counterpart does not. We provide a numerical illustration of the correlation dynamics.