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Linear and nonlinear stability analyses are performed to determine critical Rayleigh numbers ( $${Ra}_{cr}$$ ) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The $${Ra}_{cr}$$ value depends on the non-dimensional frequency $$\omega$$ of the boundary heat-flux modulation. Floquet theory is used to find $${Ra}_{cr}$$ for linear stability, and the energy method is used to find $${Ra}_{cr}$$ for two different types of nonlinear stability: strong and asymptotic. The most unstable linear mode alternates between synchronous and subharmonic frequencies at low $$\omega$$ , with only the latter at large $$\omega$$ . For a given frequency, the linear stability $${Ra}_{cr}$$ is generally higher than the nonlinear stability $${Ra}_{cr}$$ , as expected. For large $$\omega$$ , $${Ra}_{cr} \omega ^{-2}$$ approaches an $O(10)$ constant for linear stability but zero for nonlinear stability. Hence the domain for subcritical instability becomes increasingly large with increasing $$\omega$$ . The same conclusion is reached for decreasing Prandtl number. Changing temperature and/or velocity boundary conditions at the modulated or non-modulated plate leads to the same conclusions. These stability results are confirmed by selected direct numerical simulations of the initial value problem.