  We build a holographic $s$-wave conductor/superconductor model and an
insulator/superconductor model in the four-dimensional conformal anomaly
corrected~(CAC) AdS gravity. The effects of CAC parameter $\alpha$ are studied
using both numerical and analytical methods in the probe approximation.
Concretely, when the CAC parameter increases, the critical temperature
increases for the conductor/superconductor phase transition, while the critical
chemical potential decreases for the insulator/superconductor case, which
suggests that the increasing CAC parameter enhances both superconducting phase
transitions. Meanwhile, below the critical temperature or beyond the critical
chemical potential, the scalar hair begins to condense, and the condensed
phases are found to be thermodynamically stable. The critical behaviors
obtained from numerics are confirmed by our analytical analysis. For the
parameters we are considering, the energy gap in the conductor/superconductor
model decreases monotonically by increasing the CAC parameter, while for the
insulator/superconductor model the energy of quasiparticle excitations
decreases with the CAC parameter.
