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During the first three observing runs of the Advanced gravitational-wave detector network, the LIGO/Virgo collaboration detected several black hole binary (BHBH) mergers. As the population of detected BHBH mergers grows, it will become possible to constrain different channels for their formation. Here we consider the chemically homogeneous evolution (CHE) channel in close binaries, by performing population synthesis simulations that combine realistic binary models with detailed cosmological calculations of the chemical and star-formation history of the Universe. This allows us to constrain population properties, as well as cosmological and aLIGO/aVirgo detection rates of BHBH mergers formed through this pathway. We predict a BHBH merger rate at redshift zero of $5.8 \, \textrm{Gpc}^{-3} \textrm{yr}^{-1}$ through the CHE channel, to be compared with aLIGO/aVirgo's measured rate of ${53.2}_{-28.2}^{+55.8} \, \text{Gpc}^{-3} \text{yr}^{-1}$, and find that eventual merger systems have BH masses in the range $17 - 43 \, \textrm{M}_{\odot}$ below the pair-instability supernova (PISN) gap, and $>124 \, \textrm{M}_{\odot}$ above the PISN gap. We investigate effects of momentum kicks during black hole formation, and calculate cosmological and magnitude limited PISN rates. We also study the effects of high-redshift deviations in the star formation rate. We find that momentum kicks tend to increase delay times of BHBH systems, and our magnitude limited PISN rate estimates indicate that current deep surveys should be able to detect such events. Lastly, we find that our cosmological merger rate estimates change by at most $\sim 8\%$ for mild deviations of the star formation rate in the early Universe, and by up to $\sim 40\%$ for extreme deviations.