Glad you asked, for I have thought a lot about this over the past few years. Yes, with me it does change with HC and the times. In no particular order, for me they are:
1. Historical Rival: Tennessee, 3rd Saturday in October, Gen. Neyland, and all that.
2. Who Rules the Roost Rival: Florida, since Spurrier.
3. Circumstance Rival: LSU, because of the Saban connection.
4. Three-hundred and Sixty-five Days Rival. This one is the most important to me, since I have to live with these goobers the whole year round. If I lived in Tennessee, Florida or Louisana, it would be one of those.