Using a systematic perturbation expansion for slightly supercritical driving, the equation for the slow spatiotemporal variation of the amplitude of convective rolls is derived when the temperatures of the horizontal fluid boundaries are modulated harmonically in time. The modulation frequencies are considered to be large compared with the amplitude growth rate, but small enough to avoid the generation of thermal Stokes layers. The marginally stable, time-dependent linear functions determining the fast spatiotemporal periodic variations of the convective fields as well as all coefficients entering into the amplitude equation depend on the modulation and the Prandtl number. Analytic expressions for all these quantities are given for small modulation amplitudes and idealized horizontal boundary conditions. The time-dependent convective heat current is compared with previous results from few-mode Galerkin approximations.