Giant Star

After a star has exhausted its supply of hydrogen, it can undergo a number of deaths depending on its mass. All but the tiniest of stars (red dwarfs) eventually become giant stars, while the largest stars may become super-giants or even hyper-giants.

Evolution
Red Dwarfs (M and very late K stars) of less than 0.5 M☉ will stay on the main sequence for many years (sometimes for trillions of years) and as such, only hypotheses exist concerning their evolution. Theories suggest that once the red dwarf ceases to produce energy through hydrogen fusion, it will be unable to fuse helium and thus will slowly collapse into a white dwarf.

Stars of a larger mass range (between 0.5 M☉ and 8 M☉) begin to fuse helium into heavier elements (such as carbon, oxygen, neon, etc.). When this happens, their radius expands and they shine brighter, but also cool off. These red giants, once they can no longer fuse elements at their core (this stops at carbon and oxygen) expel much of their mass as a planetary nebula, while the core remains – becoming a white dwarf.

Stars of a much larger mass range (≥ 8 M☉) continue fusing elements, swelling into super-giants or hyper-giants, until at last silicon is fused into iron. When no more fusion can take place, the star explodes in a spectacular supernova, leaving behind either a neutron star or a black hole.

Around 6.6% of stars in the Milky Way are either hyper-giants, super-giants, bright giants, giants or sub-giants.

Properties and Habitability
Modelling these yourself is no simple task, but fortunately, the internet comes prepared with this stellar evolution simulator  and this Hertzsprung-Russel Diagram Explorer.

Because giant post-main sequence stars are unstable and change their radius and luminosity relatively quickly, the habitable zone (determined by the equations given above) will also be changing relatively quickly, and any one planet will not stay in the habitable zone long enough for life to evolve naturally.

Worldbuilding in Practice
"One of the many systems that will be accessed in the foreseeable future is the red giant Clytemnestra. It is a 406 million year-old star, and is about 1 million years away from its eventual death as a planetary nebula and a white dwarf. Despite its relatively small mass of 2.4 M☉, it has a radius of about 52 R☉ and is about 1,800 times as luminous as the Sun, but with a meager temperature of just 4,977 K. Despite its unwieldy size and relatively near death, it is a source of immense interest among scientists and governments alike, as evidence strongly suggests it is already colonized. No records of its colonization exist in the Galactic Union archives and no known gates link any systems to Clytemnestra, leading many to believe that it is inhabited by a space-faring species yet to be encountered by the galactic community."

"Hrimnir is a massive blue giant which sports a mineral-rich planetary system colonized by  ESIC  (the  E xtra- S olar  I nvestment  C orporation), which greatly exploits the wealth of the system. Hrimnir itself is over 70 times as massive as the Sun and is about 211 times as big as the Sun. In other words, its radius is about the size of Earth’s orbit. It is a sweltering 13,000 K and has a luminosity of  1,402,814 L☉! Even from Hyrja (a giant planet orbiting at a staggering 1,437 AU), it appears 69 times brighter than the Sun from Earth."