  We construct and analyze holographic superconductors with generalized higher
derivative couplings, in single R-charged black hole backgrounds in four and
five dimensions. These systems, which we call very general holographic
superconductors, have multiple tuning parameters and are shown to exhibit a
rich phase structure. We establish the phase diagram numerically as well as by
computing the free energy, and then validated the results by calculating the
entanglement entropy for these systems. The entanglement entropy is shown to be
a perfect indicator of the phase diagram. The differences in the nature of the
entanglement entropy in R-charged backgrounds compared to the AdS-Schwarzschild
cases are pointed out. We also compute the analogue of the entangling
temperature for a subclass of these systems and compare the results with
non-hairy backgrounds.
