Serve without too much delay. If your dish is not ready, leave the sauce to wait in the bain-marie, not too hot, with a lid.
Remarks
Some technical words: Béarnaise is a sauce of the warm emulsified kind (like hollandaise and mousseline), which means a hot base, here a well-flavoured sabayon, where butter is added to get this smooth texture.
Usually chefs make bearnaise in a different way: everything in a single pan, without a bain-marie. Faster, but more difficult.