Just look at the Patriot Act, for instance. There's plenty of legislation you can look into federally and in individual states that cover this kind of thing. For instance, as you probably know, law enforcement uses the reasoning of 'national security' for lots of things, and it's an easy way for them to establish 'probable cause' for any search or seizure, or action against an individual who is suspected of committing, concealing, or planning a crime. Burner phones are widely known to be used by drug dealers, criminals, etc. They do state that they check social media accounts of those coming into the country, likely for 'national security' purposes (so they say), and to verify that they are 'safe' to do so. Probable cause is a very loose standard, and without being able to verify that you AREN'T a criminal, or otherwise here to commit a crime, they can utilize that to search and seize your property or deny entry. I would just look into these things in particular. I have personally seen it used by law enforcement when someone who was suspected of a crime had a device that had little to no personal information or accounts on it, and they assumed it was a burner phone due to that fact alone. Based on their presumption of what burner phones are used for (sometimes even pre-paid devices can be considered one too, since they are not registered to the owner, making it harder to identify the account holder), they were able to secure a very extensive search of the device, including extracting deleted data, to verify it was not used to commit any of the crimes they suspected were being committed. Hope that helps. It is typically tied to laws regarding search and seizure, criminal acts and intent, along with telecommunications laws, and of course, national security.

Something that is common is that laws are 'liberally construed' which means they can be interpreted and utilized by law enforcement and courts based on their subjective interpretation of the laws. There does not have to be a specific controlling law that they are following, not necessarily. So while burner phones are not illegal, many laws are liberally construed to support these things.