The Sun was an order of magnitude more luminous during the first few hundred thousand years of its existence, due in part to the gravitational energy released by material accreting from the solar nebula. If Jupiter was already near its present mass, the planet's tides opened an optically thin gap in the nebula. Using Monte Carlo radiative transfer calculations, we show that sunlight absorbed by the nebula and re-radiated into the gap raised temperatures well above the sublimation threshold for water ice, with potentially drastic consequences for the icy bodies in Jupiter's feeding zone. Bodies up to a meter in size were vaporized within a single orbit if the planet was near its present location during this early epoch. Dust particles lost their ice mantles, and planetesimals were partially to fully devolatilized, depending on their size. Scenarios in which Jupiter formed promptly, such as those involving a gravitational instability of the massive early nebula, must cope with the high temperatures. Enriching Jupiter in the noble gases through delivery trapped in clathrate hydrates will be more difficult, but might be achieved by either forming the planet much farther from the star or capturing planetesimals at later epochs. The hot gap resulting from an early origin for Jupiter also would affect the surface compositions of any primordial Trojan asteroids.