  We generalize the Einstein Maxwell gravity by an additional term which is
made by contraction of the Ricci tensor and the electromagnetic antisymmetric
Maxwell tensor field, to study thermodynamic of a modified
Reissner-Nordstr\"{o}m (RN) black hole. Our motivation in use of this exotic
form of the gravity theory instead of the ordinary EM gravity is based on
influence of cosmic magnetic field throughout the spacetime as inevitable
effect at duration of cosmic inflation. While in ordinary models of
inflationary cosmology the dark energy sector is just considered. Dynamical
field equations are nonlinear and so they have not exact analytic solutions in
general form. On the other side to find equation of state of a back hole by
horizon equation we need an analytic form of the metric field solutions. Hence
we use dynamical system approach to obtain analytic metric solutions near the
critical points of secular equation of the corresponding Jacobi matrix. we
investigated stabilization situations of the obtained metric fields and finding
equations of state of modified RN AdS black hole, the Hawking temperature and
possibility of phase transitions.
