In our gravitational lifetime estimate, we assumed that the Sun shines with a constant luminosity. If this is not the case, say because it became very dim and then got brighter again, then perhaps we could stretch out the Sun’s gravitational lifetime. But this idea has major problems. First, if the Sun became dimmer by even a small amount, Earth would freeze solid. This would not be like the recent ice ages, it would be much more extreme. There is some evidence for this in the geologic record, but certainly not for multiple or extremely extended periods. For the most part, the record indicates that the Sun has shone quite steadily for the whole time Earth has existed. But even disregarding conditions on Earth, there are strong reasons to think that the Sun would not radically change its luminosity.
The Sun is a huge, gravitationally bound, ball of gas. It is governed by a condition called hydrostatic equilibrium. This condition pertains to any static fluid material. The weight of the upper layers of the fluid press down on the lower layers, and as a result, the lower layers must exert more pressure than the upper layers do, just to support the extra weight. Hydrostatic equilibrium takes its name from the pressure-balance exhibited in water, where the pressure increases linearly with depth.
Water differs in one important way from gases like those found in our atmosphere or a star. Water cannot be compressed. It stays at a near-constant density no matter what the pressure is. On the other hand, gases compress when the pressure is increased, and they expand when the pressure is decreased. In addition, as gases are compressed they heat up. It is the increase in pressure that accompanies increased temperature that supports gases in hydrostatic equilibrium.
The condition of hydrostatic equilibrium determines the Sun’s temperature. In addition, the radiation emitted by the Sun is described by the Stefan-Boltzmann law, which states that the amount of radiation emitted by an object is proportional to its temperature to the fourth power. Heating the Sun would both increase the outward pressure against gravity and cause its luminosity to rise. Both of these effects would cause the Sun to cool. So, there is a strong coupling between the energy emitted by, and the pressure supporting, the Sun. Changing either one of them would affect the other in such a way as to reverse the original change. This is called a stable equilibrium, and it suggests quite strongly that the Sun is not able to change the amount of radiation it emits without undergoing very strong structural changes, and vice versa.