  Hyperbolic plasmons are collective electron excitations in layered
conductors. They are of relevance to a number of superconducting materials,
including the cuprates and layered hyperbolic metamaterials [V. N.
Smolyaninova, et al. Scientific Reports 6, 34140 (2016)]. This work studies how
the unusual dispersion of hyperbolic plasmons affects Cooped pairing. We use
the Migdal-Eliashberg equations, which are solved both numerically and
analytically using the Grabowski-Sham approximation, with consistent results.
We do not find evidence for plasmon-mediated pairing within a reasonable
parameter range. However, it is shown that the hyperbolic plasmons can
significantly reduce the effects of Coulomb repulsion in the Cooper channel
leading to an enhancement of the transition temperature originating from other
pairing mechanisms. In the model of a hyperbolic material composed of identical
layers, we find this enhancement to be the strongest in the $d$-wave channel.
We also discuss strategies for engineering an optimal hyperbolic plasmon
background for a further enhancement of superconductivity in both $s$-wave and
$d$-wave channels.
