The state filed an appeal asking that U.S. District Judge Susan Bolton's preliminary injunction be lifted so that all of the law can take effect.
Lawyers for Gov. Jan Brewer also are asking that the appeal be considered quickly.
The judge's temporary injunction against the guts of the law didn't stop two Phoenix climbers from dangling for hours on Wednesday night, hanging off an 11-story-high crane. There were demonstrators at various sites around Phoenix, and busloads of activists from L.A. were expected to join the Phoenix protests.
Hundreds of people gathered at the door of the county jail, beating on the metal door, and at least 32 people were arrested, including a photographer for the Arizona Republic.
Dozens of others were arrested throughout the day, trying to cross a police line, entering closed-off areas or sitting in the street and refusing to leave. Former state Sen. Alfredo Gutierrez, who ran unsuccessfully for governor in 2002, was among them.
Marchers chanted "Sheriff Joe, we are here, we will not live in fear," and among the crowd was a drummer wearing a papier-mache Sheriff Joe head and dressed in prison garb.
Maricopa Sheriff Joe Arpaio announced an immigration crackdown for Thursday afternoon, saying his deputies will continue doing what they've always done - arrest illegal immigrants for crimes, find out if they're illegal afterwards, and then have them deport.
On the Capitol lawn, Phoenix resident Travis Franklin marched in support of tougher immigration laws.
"Sometimes I hear them say they haven't broken a law when, you know, the law says they have," Franklin said. "I can break the law, too, but I still got to pay the price for what I do. Everybody's got to pay the price."
Most of the marchers against the now-suspended law know the fight is not over. It's only a preliminary injunction, and it could change.
"I'm sure they're going to appeal these all the way up to the Supreme Court," said former UCLA professor Carlos Velez-Ibanez, who is now a Phoenix resident.
Others feel the same way - that they can't let down their guard.
San Diego resident Israel Puentes traveled to Phoenix to join the demonstrations.
"People who are from the state - and I lived here for three years - were under some oppression from the sheriff before this bill was signed," Puentes said.
In Los Angeles, a demonstration brought traffic to a halt in the Mid-Wilshire district on Thursday morning. About 100 members of the group We Are All Arizona gathered in the middle of Wilshire Boulevard and Highland Avenue and chained their hands together in a circle.
Los Angeles police stood by, but they have not made any arrests.
The Associated Press contributed to this story.
