Bought Under Appraised Value

I'm not sure "recoup" is the right word, because it doesn't sound as if you want to get something back that you spent.  But if you're hoping to refinance for a higher amount and come out with cash in your pocket -- the only problem might be in the rare case where your current mortgage has a penalty for paying it off too soon. 

Remember, you might come out with cash, but it's borrowed money.  No only will you be paying more interest on a larger debt, but you'll have the expense of a new set of closing costs as well.  And of course, a lot would depend on your proposed new interest rate, as compared with the one you have now.