was terminated from the job and still got paychecks for four pay periods. what is my responsibility?
I have no authority to tell you what to do.
Upon receiving the first check, I'd have contacted the payroll department of my former employer to advise them of the check, and to find out why I received it.
It could be a mistake,or it could be monies owed to you, or gifted to you.
Nevertheless, always seek an explanation, and attempt to return errant property to its correct owner.
I had a similar incident happen to me, many years ago, after retiring from the military and becoming an associate at a large law firm. I left that firm to start my own.
I inquired and was told the partners had decided to keep paying me for six more months, conditioned on me asking "why", and agreeing to do work for them during the six months PRN. It was a cozy little arrangement that went on for 24 months in total.