  Recently, a new cosmological framework, dubbed Ricci Cosmology, has been
proposed. Such a framework has emerged from the study of relativistic dynamics
of fluids out of equilibrium in a curved background and is characterised by the
presence of deviations from the equilibrium pressure in the energy-momentum
tensor which are due to linear terms in the Ricci scalar and the Ricci tensor.
The coefficients in front of such terms are called the second order transport
coefficients and they parametrise the fluid response to the pressure terms
arising from the spacetime curvature.
  Under the preliminary assumption that the second order transport coefficients
are constant, we find the simplest solution of Ricci cosmology in which the
presence of pressure terms causes a departure from the perfect fluid redshift
scaling for matter components filling the Universe. In order to test the
viability of this solution, we make four different ans\"{a}tze on the transport
coefficients, giving rise to four different cases of our model. On the physical
ground of the second law of thermodynamics for fluids with non-equilibrium
pressure, we find some theoretical bounds (priors) on the parameters of the
models. Our main concern is then the check of each of the case against the
standard set of cosmological data in order to obtain the observational bounds
on the second order transport coefficients. We find those bounds also realising
that Ricci cosmology model is compatible with $\Lambda$CDM cosmology for all
the ans\"{a}tze.
