In this case, it is not water temperature that is important (I think). In your question productivity of cooling system important. As a rule, cooling systems connected to water supply network. In technical requirements for water supply networks, you can find out requirements temperature of cold water.
I am trying to find out that; is target itself causes the reddish?
so in one hand, I think when target consumed up, low temperature of direct cooling water affect separating of Ti from Target, so we have more N and less Ti in resulted TiN film and thus the reddish appeared.