  We use strong-coupling Eliashberg theory to study the competition of separate
superconducting orders at low temperatures. Specifically, we study
magnon-mediated superconductivity in a trilayer heterostructure with a thin
normal metal between two antiferromagnetic insulators. Spin-triplet $p$-wave,
spin-triplet $f$-wave, and spin-singlet $d$-wave superconducting gaps have been
predicted to occur close to the critical temperature for the superconducting
instability. The gap symmetry with the largest critical temperature depends on
parameters in the model. We confirm that the same gap symmetries appear at any
temperature below the critical temperature. Furthermore, we show that the
temperature can affect the competition between the different superconducting
orders. In addition, we consider time-reversal-symmetry-breaking, complex
linear combinations of candidate pairings, such as chiral $p$-, $f$-, and
$d$-wave gaps, as well as $p_x+if_y$-wave gaps. We find indications that some
of these time-reversal-symmetry-breaking, nodeless gaps offer a greater
condensation energy than the time-reversal symmetric gaps. This indicates that
superconducting states with spontaneously broken time-reversal symmetry and
nontrivial topology may be preferred in this system.
