No. Size is the dominant factor. Lifespan and what stages of fusion it undergoes are dependent on its size. The more massive the star, the shorter its life span. Red dwarf stars can live for trillions of years, but massive stars may live less than a billion years.

Notice that haiguise wrote "at the core temperature of the Sun." A more massive star has a higher core temperature, and thus haiguise's sentence about fusion rates would no longer apply. Fusion rates are faster at higher temperatures, and that's why more massive stars burn out faster. Notice haiguise wrote "T^4" and "T^20." Our sun is roughly 5000K. Massive stars can exceed 10000K. At twice the temperature, T^4 and T^20 imply 16x and 1,048,576x fusion rates, respectively.

Edited to add: Wikipedia has an HR diagram with labels showing lifespans for stars at different temperatures: https://commons.wikimedia.org/wiki/File:Hertzsprung-Russel_S....