Why oh why can't we have a better press corps?
A "libertarian" is somebody who prioritizes liberty: anti-immigration anti-choice Rand Paul is many things, but he is not a libertarian.
Ross Douthat simply does not know what a libertarian is:
The Principles of Rand Paul: Paul is a libertarian, certainly... a particular kind of a libertarian... culturally conservative (opposing both abortion and illegal immigration), radically noninterventionist.. views nearly everything today’s federal government does as a violation of the founding fathers’ vision. This worldview goes by many names, including “paleoconservatism,” “the old right” and “paleolibertarianism.”
Let me turn over the microphone to Lew Rockwell, who explains what Douthat thinks Rand Paul is:
Wikipedia: In January, 1990 Llewellyn H. Rockwell, Jr. published "The Case for Paleo-libertarianism" in Liberty magazine. In it he wrote that the “conservative crack-up presents an historic opportunity for the libertarian movement” to unite with conservatives but only if “libertarianism is deloused” of those who believe in “freedom from cultural norms, religion, bourgeois morality, and social authority.” Citing drug use by libertarians and the nomination of a prostitute as the California Libertarian Party candidate for lieutenant governor, Rockwell asserted that “the only way to sever libertarianism’s link with libertinism is with a cleansing debate.” Assailing alleged “hatred of western culture,” he asserted that “pornographic photography, ‘free’-thinking, chaotic painting, atonal music, deconstructionist literature, Bauhaus architecture, and modernist films have nothing in common with the libertarian political agenda - no matter how much individual libertarians may revel in them” and stated “we obey, and we ought to obey, traditions of manners and taste.” After explaining why cultural conservatives could make a better argument for liberty to the middle classes, Rockwell predicted “in the new movement, libertarians who personify the present corruption will sink to their natural level, as will the Libertarian Party, which has been their diabolic pulpit.”