In a small saucepan, bring water to a boil.
Remove from heat, and add the 13 peach tea bags.
Steep for 10 minutes.
Lightly squeeze bags to remove excess moisture before discarding them. Set peach tea concentrate aside.
Add ketchup, peach tea concentrate, brown sugar, cider vinegar, and BBQ rub to a separate small saucepan.
Simmer over medium heat for 2 minutes, until well combined. Stir often.
Reduce heat to medium-low and let the mixture simmer for 5 minutes. Set to low if the sauce is bubbling too much.
Remove the saucepan from the heat and stir in 1 tablespoon of butter until it has completely melted.
Cover pot and keep warm before using.
If making this sauce ahead of time, let it cool completely, then transfer it to an airtight container. Store it in the refrigerator for up to 1 week. Rewarm before using.