  Determining how galactic environment, especially the high gas densities and
complex dynamics in bar-fed galaxy centers, alters the star formation
efficiency (SFE) of molecular gas is critical to understanding galaxy
evolution. However, these same physical or dynamical effects also alter the
emissivity properties of CO, leading to variations in the CO-to-H$_2$
conversion factor ($\alpha_\rm{CO}$) that impact the assessment of the gas
column densities and thus of the SFE. To address such issues, we investigate
the dependence of $\alpha_\rm{CO}$ on local CO velocity dispersion at 150-pc
scales using a new set of dust-based $\alpha_\rm{CO}$ measurements, and propose
a new $\alpha_\rm{CO}$ prescription that accounts for CO emissivity variations
across galaxies. Based on this prescription, we estimate the SFE in a sample of
65 galaxies from the PHANGS-ALMA survey. We find increasing SFE towards high
surface density regions like galaxy centers, while using a constant or
metallicity-based $\alpha_\rm{CO}$ results in a more homogeneous SFE throughout
the centers and disks. Our prescription further reveals a mean molecular gas
depletion time of 700 Myr in the centers of barred galaxies, which is overall
3-4 times shorter than in non-barred galaxy centers or the disks. Across the
galaxy disks, the depletion time is consistently around 2-3 Gyr regardless of
the choice of $\alpha_\rm{CO}$ prescription. All together, our results suggest
that the high level of star formation activity in barred centers is not simply
due to an increased amount of molecular gas but also an enhanced SFE compared
to non-barred centers or disk regions.
