Ft. Lauderdale's Mayor Naugle found out that's the case the hard way.
http://www.sun-sentinel.com/local/broward/sfl-flbnaugle0729nbjul29-story.html
Jim Naugle was once viewed as virtually invincible.He was a 24-hour mayor for Fort Lauderdale who fixed potholes when people complained and championed such popular causes as fighting overdevelopment. Polls showed his approval ratings topped 70 percent — despite his penchant for shoot-from-the-hip comments that left parts of the community feeling alienated and abused.
Times may have changed in the wake of Naugle's recent statements about homosexuality, according to political insiders and campaign experts. He ignited a firestorm when he charged the city has a problem with gay sex in public restrooms, said gays are unhappy and contended the gay Stonewall Library should not be housed in a city building because its collection contained pornography.
In an unprecedented repudiation, more than 800 people, including key members of Broward County's political power structure and business community, rallied in front of City Hall last week. Organizers said they would not try to recall Naugle because it would be difficult and expensive. Still, they would like the county and city commissions to censure him and say they will closely monitor not only his comments but those who seek to follow him.
And this was in 2007.
I remember that...but IIRC Naugle, a Democrat, never backed down.