What are the consequences of stopping an anticonvulsant?

Several. Except in a life threatening situation, you should not stop anticonvulsants abruptly. The most serious reason is that you could precipitate seizures. In addition, one can also develop other withdrawal symptoms: palpitations, anxiety, or shortness of breath. Always discuss the best way to come off drug with your physician.
Start seizures again. If you abruptly stop the anticonvulsants you may start the seizures again.