Is your check engine light on? Have you plugged into your OBDII port and checked for faults? Could be a few things, maybe fueling related. The N18's (2011-2013 Cooper S) don't typically have as many issues with the HPFP as the N14 (2007-2010 Cooper S), but it does happen. Their symptoms are very similar to what you're experiencing, but we want to properly diagnose the issue before throwing parts at it, especially expensive ones.
This is for N14's, but you'll want to test your fuel system: MINI Cooper R56 Fuel Pump Testing (2007-2011) | Pelican Parts DIY Maintenance Article
. HPFP test is the last figure in the article.
Let me know what you find out.