Strelcyk et al. (2019) recently found that interaural phase discrimination in older hearing-impaired listeners was correlated with both visuospatial processing speed and interaural level discrimination. This suggests that temporal fine structure (TFS) processing relies on global processing speed and/or spatial cognition, though it is possible that, generally, complex auditory discrimination engages multiple cognitive domains. Here, 50 Veterans (mean age = 48.1, range = 30–60) with normal or near-normal hearing completed batteries of temporal processing and cognitive tests. Composite cognitive test scores reflecting processing speed/executive function (PS-EX) and working memory (WM) were obtained. Temporal processing tasks included measures of envelope (ENV; gap duration discrimination, forward masking) and TFS (frequency modulation detection, interaural phase modulation detection) processing. Bayesian hierarchical regression was used to fit psychometric functions simultaneously to all ENV and TFS tasks in all subjects. Fixed effects of PS-EX and WM on thresholds and slopes were estimated for the psychometric functions in each temporal task. In general, (i) PS-EX and WM influenced both TFS and ENV thresholds, but not slopes; and (ii) TFS thresholds were best explained by PS-EX scores while ENV thresholds were best explained by WM scores. These findings suggest a specific relation between global processing speed and TFS.