  We study the Horndeski vector-tensor theory that leads to second order
equations of motion and contains a non-minimally coupled abelian gauge vector
field. This theory is remarkably simple and consists of only 2 terms for the
vector field, namely: the standard Maxwell kinetic term and a coupling to the
dual Riemann tensor. Furthermore, the vector sector respects the U(1) gauge
symmetry and the theory contains only one free parameter, M^2, that controls
the strength of the non-minimal coupling. We explore the theory in a de Sitter
spacetime and study the presence of instabilities and show that it corresponds
to an attractor solution in the presence of the vector field. We also
investigate the cosmological evolution and stability of perturbations in a
general FLRW spacetime. We find that a sufficient condition for the absence of
ghosts is M^2>0. Moreover, we study further constraints coming from imposing
the absence of Laplacian instabilities. Finally, we study the stability of the
theory in static and spherically symmetric backgrounds (in particular,
Schwarzschild and Reissner-Nordstr\"om-de Sitter). We find that the theory,
quite generally, do have ghosts or Laplacian instabilities in regions of
spacetime where the non-minimal interaction dominates over the Maxwell term. We
also calculate the propagation speed in these spacetimes and show that
superluminality is a quite generic phenomenon in this theory.

High Energy Physics - Theory;Cosmology and Nongalactic Astrophysics;General Relativity and Quantum Cosmology