"Once in its post-main sequence phase the Hawking star becomes fully convective and slowly swells over multiple Gyr to about 10 R⊙ and about 10 L⊙ causing them

"Once in its post-main sequence phase the Hawking star becomes fully convective and slowly swells over multiple Gyr to about 10 R⊙ and about 10 L⊙ causing them to appear as a sub-subgiant star, joining a relatively sparse population on the HR diagram." and then "The resulting increase in accretion luminosity causes the inner convective region to grow outwards, mix the He core with the H-burning shell, and quench fusion." Would the aftermath of all this be observable? Once fusion is quenched wouldn't the gas cool rapidly?