How did the Sun's luminosity change during the first billion years?

A very important and complex question. First of all, as a proto star collapses to form a stable star, its luminosity decreases while its surface temperature increases. Once it reaches the so-called Main Sequence when hydrogen can burn into helium at a stable rate, the star then begins a gradual increase in luminosity. During the first 10 to 50 million years it is still settling down as a T-Tauri star as material continues to rain down upon the stellar surface, and great winds and flares erupt on the surface as magnetic fields rearrange themselves and release energy. The T-Tauri phase then slackens and shuts off once the star's photosphere and internal structure stabalize. It is expected that a long period of luminosity increase then begins within the first billion years. Various computer calculations of the evolution of the Sun suggest that it may have been significantly less luminous in the past by perhaps 50 percent or more compared to the present day. This seems to suggest that, for the Earth, the early history of the Earth was a time when its oceans were locked in deep freeze rather than being liquid. Life may have arisen around numerous hydrothermal vents such as those we see in the bottom of the Atlantic ocean today. During the last few billion years, greenhouse heating by the atmosphere may have melted the oceans and made the Earth hospitable for life. The details are still quite uncertain. We do know, however, that fossil bacterial life can be found in 3.8 billion year old rocks so at least some fraction of the Earth was not frozen.

The first billion years may have seen the Sun 1/2 as luminous as it is today, and steadily growing to its present day luminosity in the last few billion years. Whether this was a gradual or episodic change we do not know but assume for now a gradual increase.