  In this work, we calculate the one-loop self-energy corrections to the gauge
field in scalar electrodynamics modified by Lorentz-violating terms within the
framework of the standard model extension (SME). We focus on both $CPT$-even
and $CPT$-odd contributions. The kinetic part of the scalar sector contains a
$CPT$-even symmetric Lorentz-breaking tensor, and the interaction terms include
a vector contracted with the usual covariant derivative in a gauge-invariant
manner. We computed the one-loop radiative corrections using dimensional
regularization for both the $CPT$-even and $CPT$-odd cases. Additionally, we
employed the Matsubara formalism to account for finite temperature effects.
