  We consider a vector-tensor gravitational model with terms quadratic in the
Maxwell tensor derivatives, called the Bopp-Podolsky term. The gravitational
field equations of the model and the equations describing the evolution of the
vector field are obtained and their Newtonian limit is investigated. The
cosmological implications of a Bopp-Podolsky type dark energy term are
investigated for a Bianchi type I homogeneous and anisotropic geometry for two
models, corresponding to the absence and presence of the self-interacting
potential of the field, respectively. The time evolutions of the Hubble
function, of the matter energy density, of the shear scalar, of the mean
anisotropy parameter, and of the deceleration parameter, respectively, as well
as the field potentials are obtained for both cases by numerically integrating
the cosmological evolution equations. In the presence of the vector type dark
energy with quadratic terms in the Maxwell tensor derivatives, depending on the
numerical values of the model parameters, the Bianchi type I Universe
experiences a complex dynamical evolution, with the dust Universes ending in an
isotropic phase. The presence of the self-interacting potential of the vector
field significantly shortens the time interval necessary for the full
isotropization of the Universe.
