We have theoretical constraints from general relativity. If you make models in GR that are homogeneous and isotropic, you end up with a general class of models called the FLRW models.
You phrased your question in terms of a time-dependent metric, but this is technically not quite the right way to describe it. In GR, you can take flat spacetime (Minkowski space) described in the usual coordinates, subject it to a change of coordinates, and come up with a new description in which the metric varies. So instead of talking about whether an FLRW metric is time-dependent, we need to talk about time-dependence using definitions that are coordinate-independent. The coordinate-independent way of talking about it is whether or not the spacetime is static.
FLRW models are not static except in the special case where the density of matter is zero. We observe that the density of matter is not zero. (As pointed out by HeisenbergImage in a comment, you can also have the Einstein static universe, which is unstable.)
You phrased the question in terms of a time-dependent metric, which is not really a good way to define the issue. Depending on how you fix that problem, you could end up talking about the empty FLRW, Einstein static universe, the steady-state model (A.V.S.'s answer), or the vacuum dominated FLRW. None of these is an accurate description of our present or past universe, although the vacuum-dominated FLRW will eventually be a very good approximation.