Do I have to do research at the college I am attending to get into a good grad school? I am currently an incoming freshman, starting a physics/math double major at UMass Boston in the fall. Over the summer, I started an internship at the Harvard Laboratory for Particle Physics and Cosmology that I really love and intend to continue doing throughout the school year.

I'm actively contributing to the research being done and I want to continue working there as long as I can. I understand that doing some level of research during undergrad is basically required to get into grad school, but it is necessary that it be at the same school that I am attending? 

Should I be focusing more on opportunities at UMass Boston? Or does it not matter where I do the research, as long as I do some sort of research for undergrad?

Thanks in advance for any help you can give me!
In short, the answer is no, it doesn't matter where your research experience is. 

If anything, I would say that it is more beneficial to have experience outside your undergraduate institution than inside, as this gives you the opportunity to meet many more potential collaborators, exposes you to new ideas and different research cultures and will hopefully give you a wider knowledge of your subject and academia in general.

However, if you're intending to continue at your current institution as a postgraduate student, then it would be wise to try and do some kind of research project (your course may require a final year dissertation, thesis, project or similar) with a current member of staff. This means you get to know them and more importantly, they get to know you and will remember your name when your application for postgraduate studies lands on their desk. 

If a professor already knows you as a hardworking, reliable undergraduate from that summer project you did with them, they're much more likely to take you on in the expectation you will also be a hardworking, reliable postgraduate (in comparison to the stranger's application next to yours on their desk). Even if you don't intend to apply at the same institution for postgraduate studies, having a supervisor who can write you a strong reference is always a bonus.

Make the most of any research opportunity you get, and good luck with your studies (cosmology is a good choice!).