Unless it was left off of yours for some reason, there should have been a heat-resistant sleeve around it under the dash. If this is the wire you're talking about?
If the Brown wire between the starter relay and the connector near the firewall, that's not a resistor wire and should not get hot.
Presumably though, you're talking about the one under the dash. It does get hot, it should have a protective sleeve around it, and like Viper said, don't leave the key in the RUN position for any longer than you have to.
Paul