  We consider the dynamics of a barotropic cosmological fluid in an
anisotropic, Bianchi type I space-time in Eddington-inspired Born-Infeld (EiBI)
gravity. By assuming an isotropic pressure distribution, we obtain the general
solution of the field equations in an exact parametric form. The behavior of
the geometric and thermodynamic parameters of the Bianchi type I Universe is
studied, by using both analytical and numerical methods, for some classes of
high density matter, described by the stiff causal, radiation, and pressureless
fluid equations of state. In all cases the study of the models with different
equations of state can be reduced to the integration of a highly nonlinear
second order ordinary differential equation for the energy density. The time
evolution of the anisotropic Bianchi type I Universe strongly depends on the
initial values of the energy density and of the Hubble function. An important
observational parameter, the mean anisotropy parameter is also studied in
detail, and we show that for the dust filled Universe the cosmological
evolution always ends into an isotropic phase, while for high density matter
filled universes the isotropization of Bianchi type I universes is essentially
determined by the initial conditions of the energy density.
