The Christian bakers from Oregon who were ordered to pay a fine of $135,000 for refusing to make a cake for a same-sex couple have filed an appeal to overturn the ruling.

Aaron and Melissa Klein of Cakes by Melissa, through the First Liberty Institute, is asking the Oregon Court of Appeals to annul the ruling made by the Oregon Bureau of Labor and Industries (BOLI) last year, arguing that it violated their constitutional rights to religious freedom, free speech and due process.

Lawyers for the First Liberty Institute said before hearing the Kleins' case, BOLI Commissioner Brad Avakian made public comments on social media where he prejudged the case against the Kleins.

Avakian said the Kleins had "disobey[ed]" Oregon law and needed to be "rehabilitate[d]." By failing to recuse himself from the case, Avakian deprived the Kleins the right to due process, the lawyers said.

They added that the penalty against the Kleins was excessive and the ruling was against the state and U.S. Constitutions.

The case stemmed from a visit by Rachel Cryer and her mother to Sweet Cakes by Melissa on Jan. 17, 2013.

Cryer told Aaron Klein that she wanted to buy a cake for her wedding. He asked the names of the bride and groom and Cryer said it was for two women.

Klein apologised, saying that because of their family's religious beliefs they could not make a custom-designed cake for the purpose.

BOLI ruled that the Kleins violated Oregon's anti-discrimination law by refusing to make a wedding cake.

"In America, you're protected by the Constitution and you're also innocent until proven guilty," said Kelly Shackelford, president and CEO of First Liberty Institute. "Commissioner Brad Avakian decided the Kleins were guilty before he even heard their case.

"The Constitution guarantees the rights of free exercise of religion, free speech, and due process for every American," said Boyden Gray, former ambassador to the European Union and founding partner of Boyden Gray and Associates.

Meanwhile, in another religious freedom case involving a baker, the Colorado Supreme Court has declined to hear the case of Mastepiece Cake owner Jack Phillips, who was found to have violated the state's law against refusing to provide service based on sexual orientation, race, sex or marital status.

The decision upholds a ruling by the Colorado Court of Appeals, which said that Phillips cannot cite his religious beliefs or free speech to make a wedding cake for a same-sex couple, the Denver Post reports.

In 2012, Phillips declined to make a wedding cake for same-sex couple Charlie Craig and David Mullins because of his religious beliefs.

Administrative Law Judge Robert Spencer said in December 2013 that offering services to gay couples did not violate Phillips' rights.

The Colorado Civil Rights Commission later required him to submit quarterly reports for two years to show that he was working to change policies.

"We asked the Colorado Supreme Court to take this case to ensure that government understands that its duty is to protect the people's freedom to follow their beliefs personally and professionally, not force them to violate those beliefs as the price of earning a living," said Jeremy Tedesco, senior counsel at the Alliance Defending Freedom, said.