Some time ago I made this spreadsheet for main sequence stars:
https://docs.google.com/spreadsheets/d/18dC2D_xyW3tFvWT47kb7Om6hW4YaHpwtUIYtEjvPEEM/edit?usp=sharing
it relates some basic relationships of stellar characteristics and evolution. It's in many ways simplistic, but should give you some idea of what are the scales and directions of tradeoffs between the size of the habitable zone, and the stellar life time.
There could probably be some exotic exemptions to what's in there (e.g. very fast rotators), but I wouldn't bet too much money on those.
The table goes only to something like 5 solar masses, but at that stage the star is already so short-lived, that forming solid planetary crust, let alone evolving a planetary ecosystem, becomes a major stretch.
I've never looked into the lore of Firefly, so I'm not sure what are the orbits of those planets, but you should be able to ascertain whether they fit into the ranges provided in the spreadsheet.
O.k., having just now looked it up, it appears that there are many systems in the 'Verse', most of which consisting of multiple stars (see the poster shown here:
http://serenity.popapostle.com/html/episodes/Serenity.htm ) This in effect gives you many habitable zones.
newjerseyrunner said:
Would a Red Giant fit the bill?
That's already a dying star, which has been significantly changing its size for the past few millions of years. Planets need time and stability to evolve habitable conditions, and the red giant stage doesn't provide much of either.