We know that presently Sun is 4.5 billion years into its main sequence. It has another 5 billion years before it enters the Red Giant phase. We also know that Sun's luminosity increases by 10% every billion years during the main sequence. I am interested in finding the temperature rise as we approach the end of main sequence. I got two different values for temperature on Earth.
Wikipedia entry says that temperature on Earth would be 422 k in 2.8 billion years. However, if we use the formula for effective temperature as discussed in this answer
https://earthscience.stackexchange.com/a/4274/15299 and L = 1.8 then, the temperature on Earth would be 330K. Also in this book, the author does the same calculations on page 255.
The difference is that your analysis is assuming that the albedo stays fixed, so the surface temperature simply scales like luminosity to the 1/4 power. The Wiki entry is including feedback from the greenhouse effect, which will tend to further increase the surface temperature. Note that an analysis that just looks at solar irradiation would get way too low a surface temperature for Venus, for example. I can't speak to the accuracy of the feedback included-- as I understand it, that is far from a simple effect to include.