There are several anecdotal explanations for how the place received its name; some involve stories of a bear cub running. In fact in southeastern jargon a "run" is a creek, and Cub Run is a creek which is a tributary of the Green River on the northern edge of Mammoth Cave National Park, and has its origin in the village of Cub Run. The species of cub is lost in history but very likely a bear cub given the proximity of other creeks such as Bear Creekand Bar Brook.
Students in Cub Run attend Hart County Schools, including Hart County [High School (Kentucky)|Hart County High School] in Munfordville. Cub Run Elementary is located on KY 728 on the south side of the community.