Russian legislator Vitaly Milonov accused the Austrian performer of "blatant propaganda of homosexuality and spiritual decay."
"I can only say `Thank you for your attention!' If this is only about me and my person, I can live with it,'" Wurst said about her critics in emails Wednesday with The Associated Press.
"You know, I have a very thick skin. It's just strange that a little facial hair causes that much excitement. I also have to add that 80 percent of the autograph requests that I get are from Russia and eastern Europe - and that's what is important to me," she said.
Lisanne Wilken, an anthropologist at Aarhus University in Denmark, said the criticism against Wurst would have been trivial if it weren't for the Russian law last year prohibiting so-called gay "propaganda."
The European Broadcasting Union, which organizes the event, has not received any formal complaints from the participating broadcasters, spokesman Jarmo Siim said. A national broadcaster is not allowed to edit out the live coverage, according to the EBU.
Neuwirth, who was born in Gmunden, central Austria, entered show business eight years ago, taking part in an Austrian TV talent show. After joining a boy band that quickly broke up, Neuwirth first appeared as Wurst in another Austrian talent show in 2011. She also took part in two reality shows, including one where candidates had to survive in the Namibian desert together with native tribes.
As she prepared for her performance in Copenhagen, Wurst said she wasn't paying much attention to the controversy about her.
"Hey, I'm just a singer in a fabulous dress, with great hair and a beard," Wurst told AP.
The annual competition is supposed to be completely removed from politics. Neither Russia's entry - teenage twins Anastasia and Maria Tolmachevy - nor Ukraine's Mariya Yaremchuk, whose routine includes a dancer running in a giant hamster wheel, allude to the recent tensions between Moscow and Kiev.
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However, the Ukraine crisis has raised anew questions about the competition's scoring, which is partly done by phone voting across Europe.
Even though Russia has annexed Crimea, votes from the Black Sea peninsula will count for Ukraine because the phone operator there is still Ukrainian, the EBU said.
At Tuesday's semifinal, the audience booed when it was announced that the Tolmachevy twins had qualified for Saturday's final but cheered when Yaremchuk got into it too.
Both have "become the human face of the conflict," Wilken said, noting Ukraine seems to be some receiving sympathy support.
Bookmakers place Ukraine in the top 10 and Russia at the bottom.
The winner is picked by juries and television viewers across Europe. The final tally for each country is a 50/50 combination of the telephone votes and votes of a national jury. A country that received a good result in the telephone vote could still be left with no points in the overall tally if the jury gave its highest points to other contestants, according to EBU.
Contest watchers believe Wurst will advance from the second semifinal Thursday. Others considered strong contenders include Armenia's Aram MP3, who fuses a traditional piano ballad with contemporary dub step beats; a haunting melody with traditional sounds from Azerbaijan; a bluegrass tinged World War I homage from Malta and Hungary's New York-born singer Andras Kallay-Saunders, whose high-energy song about domestic violence could end up being a chart hit across the continent.
The competition is hosted by Denmark, the winner of last year's contest. Organizers say they expect 180 million television viewers this year.