  Oxygen isotopes in marine cherts have been used to infer hot oceans during
the Archean with temperatures between 60{\deg}C (333 K) and 80{\deg}C (353 K).
Such climates are challenging for the early Earth warmed by the faint young
Sun. The interpretation of the data has therefore been controversial. 1D
climate modeling inferred that such hot climates would require very high levels
of CO2 (2-6 bars). Previous carbon cycle modeling concluded that such stable
hot climates were impossible and that the carbon cycle should lead to cold
climates during the Hadean and the Archean. Here, we revisit the climate and
carbon cycle of the early Earth at 3.8 Ga using a 3D climate-carbon model. We
find that CO2 partial pressures of around 1 bar could have produced hot
climates given a low land fraction and cloud feedback effects. However, such
high CO2 partial pressures should not have been stable because of the
weathering of terrestrial and oceanic basalts, producing an efficient
stabilizing feedback. Moreover, the weathering of impact ejecta during the Late
Heavy Bombardment (LHB) would have strongly reduced the CO2 partial pressure
leading to cold climates and potentially snowball Earth events after large
impacts. Our results therefore favor cold or temperate climates with global
mean temperatures between around 8{\deg}C (281 K) and 30{\deg}C (303 K) and
with 0.1-0.36 bar of CO2 for the late Hadean and early Archean. Finally, our
model suggests that the carbon cycle was efficient for preserving clement
conditions on the early Earth without necessarily requiring any other
greenhouse gas or warming process.
