Some abilities such as Strength of the Sun give a bonus to Charisma-based checks.
On the other hand, paladins, bards, and sorcerers add their Charisma modifier to Concentration checks.
In that situation, is Concentration a "Charisma-based check" for paladins, bards, and sorcerers?
Yes, the bonus from Strength of the Sun would apply to Charisma-based concentration checks.
The trait doesn't call out specific types of checks, like skill checks or saves, so it therefore applies to all checks that use Charisma as a modifier.
Any check that is modified by Charisma as opposed to another attribute is a Charisma-based check.
RAW: 'X-based checks' are not explicitly defined anywhere.
However, I would consider concentration checks to be based off of both of the primary casting attribute of the caster. Mostly because skills are also never explicitly defined as 'ability based' but are referenced as such.
The format for a skill check is :
1d20 + Ranks +KeyAbilityMod +other mods
A skill check is considered X-based when X is they key ability used.
Concentration follows a similar check:
1d20 + CasterLevel + CastingAbilityMod + other mods
We can create this mapping between the modifiers for skill check and concentration checks:
Ranks -> caster level
KeyAbilityMod -> CastingAbiiltyMod
other mods -> other mods
If you agree with such a mapping then that leads to considering concentration to based on your casting attribute.
note: monsters in the bestiary have abilities that explicitly state when their abilities are 'x-based'. However, I think this is because they leave out the formulas to calculate the values and certain situations (ability damage) can change them.
