Prepare boiling water canner(Or large pot). Heat jars and lids in simmering water until ready for use. Do not boil. Set bands aside.
Combine peaches, brown sugar, granulated sugar, rum and lemon zest in a large stainless steel saucepan. Bring to a boil over high heat, stirring constantly, until sugar dissolves. Reduce heat and boil gently, stirring occasionally, until thickened, about 20 minutes.
Ladle hot sauce into hot jars leaving 1/4 inch headspace. Remove air bubbles and re-measure headspace. If needed, add more sauce to meet recommended headspace. Wipe rim. Center lid on jar. Apply band and adjust until fit is fingertip tight.
Process filled jars in a boiling water canner for 10 minutes, adjusting for altitude. Remove jars and cool. Check lids for seal after 24 hours. Lid should not flex up and down when center is pressed