Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Reserve 1/2 cup cooking water, then drain the pasta.
Meanwhile, pile the garlic, 1 tablespoon capers and the red pepper flakes on a cutting board; mince, then mash with the flat side of a chef's knife to make a paste. Transfer the mixture to a large bowl and add the olive oil, tomatoes, olives, the remaining 1 tablespoon capers and the butter. Stir in the basil and cheese.
Add the pasta to the bowl with the tomato mixture and toss until the butter is melted, adding the reserved cooking water to moisten, if needed. Season with salt.