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 ansätze 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 $$\Lambda $$ Λ CDM cosmology for all the ansätze.