Wash out the pot you used to blanch the peaches, then add the granulated sugar, brown sugar and 8 cups of water. Bring to a simmer over medium heat.
Meanwhile, pack the peaches into the jars, cut sides down.
Ladle the hot syrup into each jar, leaving enough room for 1/2 cup of rum.
Pour 1/2 cup rum into each jar and top off with additional syrup if necessary to leave 1/2 to 1/4 inch headspace at the top of each jar.
Wipe the rims clean with a damp paper towel. Top with a hot lid and a band.
Process in a boiling water bath for 25 minutes.
Remove jars using tongs and place on a towel-lined jelly roll pan or countertop to cool.
Over the next several hours you’ll hear a “ping” as the lids go concave signaling a secure seal.
If the lids do not go concave, refrigerate the jar and use it first.