The Tweet @-mentioned both the Lightning and the NHL in the afternoon of Oct 7th, and the suspension came a bit over 2 days after, the evening of the 9th, so it wasn't quite that quick. They were likely looking into this before it blew up in the news, though obviously it blowing up in the news would apply pressure to act:
Also, the NHL is a small, tight knit community, and the people making these suspension decisions are literally a part of that community. They're well connected to Cole's ex teammates, coaches, trainers, etc. It's not hard for them to very, very quickly get to a decent early take of "this seems like BS" to "this could be legit". If you're in Lightning management, doesn't take long tend send out a few "early feeler" texts/emails/calls to your friends who know Cole to help you decide on your first actions.
They aren't passing any final judgements off of that, there will be a proper investigation, but I'll bet they have a lot more information than the general public does. That doesn't mean Cole is definitely guilty, but it does mean not all accusations will result in immediate suspensions.