It depends on what exactly information he received. Knowledge on the location of the fragments of the blade might lead to him recovering them, and then promptly turning them over to the Ebon Blade. He feels it was foolish to leave something so important in the hands of the Argent Crusade, given how inept they have proven themselves to be with runeblades before. It’s also not something that should just be left ‘lost’ since there is still the potential for it falling into the wrong hands. He’d also prefer not to be targeted for having it. (It looks goddamn silly on a blood elf anyway. It’s like a toothpick.)
Information on Frostmourne’s powers would be far more useful. It would be a lot less risky for him to its abilities into his own runeblade, and the combat advantage it would provide would be invaluable.