I am 5 months pregnant. Is the father of my unborn child financially responsible for me and the unborn baby?

Only if you are married to him.

Since you are not married no because you have to establish paternity before he is financially responsible. So you have to wait until the baby is born and he signs the birth certificate. He can request a paternity test. Then you go to court and file for child support
You can talk to him about helping financially for your pregnancy and if he does that is great.

Not in the U.S. he isn't. He isn't responsible until paternaty is established (assuming you are not married to him). Paternaty can be established one of two ways... either he also believes to be the father and he signs the birth certificate or a judge has to order a DNA test.