We consider a colloidal particle immersed in an active bath and derive a Smoluchowski equation that governs the dynamics of the colloidal particle. We address this as active Smoluchowski equation. Our analysis based on this active Smoluchowski equation shows a short time superdiffusive behavior that strongly depends on the activity. Our model also predicts a non-monotonic dependence of the mean energy dissipation against time, a signature of activity-induced dynamics. By introducing a frequency-dependent effective temperature, we show that the mean rate of entropy production is time-dependent, unlike in a thermal system. The main reason for these anomalies is the absence of any fluctuation–dissipation theorem for the active noise. We also comment on how microscopic details of activity can reverse the trends for the mean energy dissipation and mean rate of entropy production. •A Smoluchowski equation that governs the dynamics of a colloidal particle in an active bath is derived.•Frequency-dependent effective temperature has been used to quantify entropy production.•Quantification of energy dissipation has been made using the Harada-Sasa equality and force-position correlation.