Assuming that you use summon greater demon (Xanathar's Guide to Everything, p. 166), and on your next turn, you still have the "control" over him granted by summon greater demon.
Now I want to use charm monster (Xanathar's Guide to Everything, p. 151) on the demon, so that (assuming I succeed) if I lose my concentration on the summon greater demon spell, I still have a demon by my side for 1d6 rounds.
Does the demon have advantage on the saving throw against charm monster due to the fact that it's hostile to me?
Best Answer
It is hostile, but that will not affect charm monster until you lose control
My other answer makes the assumption that hostile means the same thing as fighting against you, but this isn't necessarily true. Taking from this answer to a different question about what hostile means, the DMG (pg. 244) describes the following:
Since a hostile creature "doesn't necessarily attack them on sight", this means that a demon doesn't have to be fighting the party to be considered hostile, and clearly demons are hostile to everything with regards to opposing everyone's goals.
Despite being hostile, the demon isn't fighting against you initially because the summon greater demon spells allows you to maintain control over the demon until it breaks free, after which point it attacks any non-demon, including you, the caster.
Whilst it isn't free of your control, it will not fight you unless you fight it first, so it will not give the demon advantage against charm monster for that reason, since that spell specifically says "fighting".
Charm monster (Xanathar's Guide to Everything, pg. 151) says:
Also note that, as described in this question, some demons have Magic Resistance, meaning those demons will have advantage against charm monster anyway, regardless of whether you're fighting it or not, so although it's hostile, and you may or may not be fighting it, it'll still have advantage on the saving throw.
Also note that having used its true name as part of the summoning spell only affects the summoning spell, so it'll always have advantage against charm monster, being a different spell.