  By keeping the cosmological principle i.e., an isotropic and homogeneous
universe, we consider the cosmology of a vector-tensor theory of gravitation
known as the \textit{bumblebee} model. In this model a single Lorentz-violating
timelike vector field with a nonzero vacuum expectation value (VEV) couples to
the Ricci tensor and scalar, as well. Taking the ansatz $B(t)\sim t^\beta$ for
the time evolution of the vector field we derive the relevant dynamic equations
of the Universe, where $\beta$ is a free parameter. In particular, by employing
observational data coming from the Big Bang Nucleosynthesis (BBN) and the
matter-antimatter asymmetry in the Baryogenesis era, we impose some constraints
on the VEV of the bumblebee timelike vector field i.e., $\xi b^2$, and the
exponent parameter $\beta$. The former and the latter limit the size of Lorentz
violation, and the rate of the time evolution of the background
Lorentz-violating bumblebee field, respectively.
