  The population of optical transients evolving within a time-scale of a few
hours or a day (so-called fast optical transients, FOTs) has recently been
debated extensively. In particular, our understanding of extragalactic FOTs and
their rates is limited. We present a search for extragalactic FOTs with the
Tomo-e Gozen high-cadence survey. Using the data taken from 2019 August to 2022
June, we obtain 113 FOT candidates. Through light curve analysis and
cross-matching with other survey data, we find that most of these candidates
are in fact supernovae, variable quasars, and Galactic dwarf novae, that were
partially observed around their peak brightness. We find no promising candidate
of extragalactic FOTs. From this non-detection, we obtain upper limits on the
event rate of extragalactic FOTs as a function of their time-scale. For a very
luminous event (absolute magnitude $M<-26$ mag), we obtain the upper limits of
$4.4 \times 10^{-9}$ Mpc$^{-3}$ yr$^{-1}$ for a time-scale of 4 h, and $7.4
\times 10^{-10}$ Mpc$^{-3}$ yr$^{-1}$ for a time-scale of 1 d. Thanks to our
wide (although shallow) surveying strategy, our data are less affected by the
cosmological effects, and thus, give one of the more stringent limits to the
event rate of intrinsically luminous transients with a time-scale of $< 1$ d.
