I'm not sure that there are any universities where you can major in cosmology (and even if there are, I'm not sure that they will be advantageous: it's best to get a strong undergrad knowledge of physics/maths and then specialise later).
Realistically, it doesn't matter much on what Undergraduate university you attend. As long as it has a physics program that interests you and is a strong program.
Also, if you are really interested in cosmology, try looking for universities where they have people working in cosmology (or a related-field). Places like UC Irvine have a group of people who are working in this field under the name of Center for Cosmology, Department of Physics & Astronomy. It is really important to be in a place with cosmologists since you want to include undergraduate research in cosmology on your CV when applying to grad school (if that is the path you wish to take).
My main advice is: (1) pick a university with a suitable physics program; (2) try to pick one that has cosmologists working there (not necessarily having an academic cosmology program).
We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling We Value Civility
• Positive and compassionate attitudes
• Patience while debating We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving