  We consider the equivalence problem for cosmological models in
four-dimensional gravity theories. A cosmological model is considered as a
triple $(M, {\bf g},{\bf u})$ consisting of a spacetime $(M, {\bf g})$ and a
preferred normalized time-like vector field ${\bf u}$ tangent to a congruence
of fundamental observers. We introduce a modification of the Cartan-Karlhede
algorithm by restricting to frames adapted to ${\bf u}$ and including the
covariant derivatives of ${\bf u}$ along with the Riemann tensor and its
covariant derivatives. To fix the frame we make use of quantities relative to
the fundamental observers, such as the anisotropic pressure tensor, energy flux
vector, electric and magnetic parts of the Weyl tensor and the kinematical
quantities of ${\bf u}$. This provides a simpler way to construct a list of
invariants relative to the fundamental observers that completely characterizes
the model, independent of coordinates. As an illustration of the algorithm, we
consider several well-known cosmological models from General Relativity.
