This is a complicated question.  Cox proportional hazards regression is used when the outcome is a binary failure (yes/no) at a given time for multiple objects (ball bearing sets, research animals, human subjects, etc).  You have three possible outcomes as a function of time.  You could use each level of your categorical outcome (0,1,2) as a yes/no failure as a function of time, and run failure analysis (Cox PH) on each outcome separately -- otherwise, you are throwing everything into a hopper and saying that you want to analyze everything in one step.  

The time interval has to be uniform, so say, for each 5-minute window for your level=0 outcome (number of double fouls = 6 per game), you code the result as a 1-failure (it did happen) or it didn't (code 0), and log the time at which the failure occurred.  You also need a list of essentially all the e.g. 5-minute intervals and a yes/no for failure.  Then input the 2 outcomes (failure (y/n) and time) into Cox PH.  

There are some concerns that you are only looking at 1 research subject, since this can affect correlation assumptions for Cox PH.  

Overall, I would suggest talking with a sports medicine researcher at at local university -- since they would know how to do this more specifically.  Thus, your question is a sports medicine question that only focuses on one subject, not a generic "textbook" statistical analysis question which commonly is based on a sample of athletes.