Motivated by recent experiments, we investigate the pressure-dependent electronic structure and electron-phonon (e-ph) coupling for simple cubic phosphorus by performing first-principles calculations within the full potential linearized augmented plane-wave method. As a function of increasing pressure, our calculations show a valley feature in T_c, followed by an eventual decrease for higher pressures. We demonstrate that this T_c valley at low pressures is due to two nearby Lifshitz transitions, as we analyze the band-resolved contributions to the e-ph coupling. Below the first Lifshitz transition, the phonon hardening and shrinking of the γ Fermi surface with s-orbital character results in a decreased T_c with increasing pressure. After the second Lifshitz transition, the appearance of δ Fermi surfaces with 3d-orbital character generate strong e-ph interband couplings in αδ and βδ channels, and hence lead to an increase of T_c. For higher pressures, the phonon hardening finally dominates, and T_c decreases again. Our study reveals that the intriguing T_c valley discovered in experiment can be attributed to Lifshitz transitions, while the plateau of T_c detected at intermediate pressures appears to be beyond the scope of our analysis. This strongly suggests that aside from e-ph coupling, electronic correlations along with plasmonic contributions may be relevant for simple cubic phosphorus. Our findings hint at the notion that increasing pressure can shift the low-energy orbital weight towards d character, and as such even trigger an enhanced importance of orbital-selective electronic correlations despite an increase of the overall bandwidth.