You definitely don't want to do that. You're going to be overwhelmed with support requests (for an e-book, even!) from the least competent potential users imaginable. Moreover, you're going to blunt the reason for writing the book in the first place, because the people best positioned to get value from the book will not do so because it is too cheap to be taken seriously as business advice.
A $100+ CPA is also a great way to think about it. I think you could get some decent affiliate traffic (which is what this is, really) for half that.
Side note: I visited App Sumo's site as a non member and there's nothing to tell me what the site is, or why I should sign up, other than it's a "store for entrepreneurs." Presumably they expect you to Google what it is, or perhaps more likely, arrive having already been explained elsewhere what it is and decided you want in. But it seems strange they wouldn't want more control over the message. I'm assuming this works for them, but I'm not sure why.
I understand that there is no variable cost, but by offering a permanent "discount", OP may have permanently devalued his product. If that steep of a cut was viable, then author might have had a bad price point to begin with.
In general, the same thing must be considered for products with little to no variable costs. I'm not sure what the optimal solution is, but I would think that one would want to sell even at $0.01 - because a sale on a product with no variable cost has to be better than no sale at all...