The auditory steady state response (ASSR) arises when periodic sounds evoke stable responses in auditory networks that reflect the acoustic characteristics of the stimuli, such as the amplitude of the sound envelope. Larger for some stimulus rates than others, the ASSR in the human electroencephalogram (EEG) is notably maximal for sounds modulated in amplitude at 40 Hz. To investigate the local circuit underpinnings of the large ASSR to 40 Hz amplitude-modulated (AM) sounds, we acquired skull EEG and local field potential (LFP) recordings from primary auditory cortex (A1) in the rat during the presentation of 20, 30, 40, 50, and 80 Hz AM tones. 40 Hz AM tones elicited the largest ASSR from the EEG acquired above auditory cortex and the LFP acquired from each cortical layer in A1. The large ASSR in the EEG to 40 Hz AM tones was not due to larger instantaneous amplitude of the signals or to greater phase alignment of the LFP across the cortical layers. Instead, it resulted from decreased latency variability (or enhanced temporal consistency) of the 40 Hz response. Statistical models indicate the EEG signal was best predicted by LFPs in either the most superficial or deep cortical layers, suggesting deep layer coordinators of the ASSR. Overall, our results indicate that the recruitment of non-uniform but more temporally consistent responses across A1 layers underlie the larger ASSR to amplitude-modulated tones at 40 Hz.