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The spell Armor of Agathys states:
You gain 5 temporary hit points for the duration. If a creature hits you with a melee attack while you have these hit points, the creature takes 5 cold damage.
Now, assume my buddy uses Inspiring Leader to give me 10 temporary hit points — does the effect end?
From the Player's Basic Rules, page 76:
If you have temporary hit points and receive more of them, you decide whether to keep the ones you have or to gain the new ones.
So in this situation you can choose to have 5 temporary hit points or 10 - it's up to you.
Armor of Agathys specifies that its effect of damaging creatures that attack you happens 'while you have these hit points'. If you replace the hit points it gave you with different hit points, you don't 'have these hit points' any more, you have some other set of hit points, so you wouldn't get the damage effect.
Cases like this are presumably why you get to choose between temporary hit points instead of automatically getting the larger amount.
Armor of Agathys PHB p.215
You gain 5 temporary hit points for the duration. If a creature hits you with a melee attack while you have these hit points, the creature takes 5 cold damage.
The spell clearly states that the creature hitting you takes damage while you have the temporary hit points granted by the spell. If they are all used up or your replace them with those from another source then they are no longer "these" hit points so the creature takes no damage.
Interestingly it does not say the spell ends at this point so, RAW, it carries on running for the full 1 hour duration.
You replace the hit-points with the new ones if you so choose. Unless it is Arcane Ward which is not actually THP, it is a separate entity that takes damage for you. It will also stack with HP gained by Polymorph which behaves like THP but also is not. See page 115 of the PHB, under Arcane Ward. Also, with THP you use the highest hit points; if you have 5 THP from armor of Agathys and false life is cast on you for 7 THP, you must decide whether or not you have and you 5 THP or 7 THP; if you opt for the 7, armor of Agathys ends. Oddly enough, false life does not end when its THP runs out; see PHB 198 for the rules on temporary hit points.
