1. What does it mean if your relationship with someone has become "cold"?

When your relationship has become "cold " it would have to mean that it's probably not the same as how it was in the beginning. also it can mean differently if you're talking about a friendship with someone OR a relationship.

In regards to history, a cold relationship can mean a state of military tension between countries. For example, the cold war was a state of military tension, where the United States and the Soviet Union kept trying to one up each other with military power. However, the cold war never amounted to anything - it was a time of tension and threats with no actual military action.