We experimentally study the transient voltage response of yttrium iron garnet/platinum bilayer samples subject to periodic heating up to gigahertz frequencies. We observe an intrinsic cutoff frequency of the induced thermopower voltage, which characteristically depends on the thickness of the yttrium iron garnet film. The cutoff frequency reaches values of up to 350 MHz in a 50 nm thick yttrium iron garnet film, but drops to below 1 MHz for several-micrometer-thick films. These data corroborate the notion that the magnon spectrum responsible for the spin current emission in the spin Seebeck effect can be shaped by tuning the thickness of the ferromagnetic layer.