We present measurements of the azimuthal rotation velocity $\dot{{\it\theta}}(t)$ and thermal amplitude ${\itδ}(t)$ of the large-scale circulation in turbulent Rayleigh–Bénard convection with modulated rotation. Both $\dot{{\it\theta}}(t)$ and ${\itδ}(t)$ exhibit clear oscillations at the modulation frequency ${\itω}$. Fluid acceleration driven by oscillating Coriolis force causes an increasing phase lag in $\dot{{\it\theta}}(t)$ when ${\itω}$ increases. The applied modulation produces oscillatory boundary layers and the resulting time-varying viscous drag modifies ${\itδ}(t)$ periodically. Oscillation of $\dot{{\it\theta}}(t)$ with maximum amplitude occurs at a finite modulation frequency ${\itω}^{* }$. Such a resonance-like phenomenon is interpreted as a result of optimal coupling of ${\itδ}(t)$ to the modulated rotation velocity. We show that an extended large-scale circulation model with a relaxation time for ${\itδ}(t)$ in response to the modulated rotation provides predictions in close agreement with the experimental results.