A district court judge failed to keep an open mind and give meaningful consideration to Kim Dotcom's arguments against extradition, defence lawyers say.
Mr Dotcom and his three co-accused - Bram Van der Kolk, Matthias Ortmann and Finn Batato - have appealed against a North Shore District Court judgement ruling them eligible for extradition to the United States.
The US wants to extradite the men to face criminal charges of money-laundering and copyright breaches related to the defunct file-sharing website Megaupload.
Crown lawyers have argued on behalf of the US that Megaupload was a "simple scheme of fraud" that wilfully breached copyright on a huge scale by hosting illegally-created movie, music and software files. The site aided and abetted users who uploaded popular illegal files by paying them financial rewards, the US said.
Mr Dotcom's lawyers argued Megaupload was set up as a legitimate file-storing site, and there were a large number of people who were using it for that purpose. They said their client could not be held responsible for the illegal actions of some users.
The four men are also appealing the district court's decision to throw out their application for a stay, which would have halted extradition proceedings indefinitely.
They asked for the stay because of what they argued was unlawful behaviour by the US before and after their arrests in 2012.
Mr Van der Kolk and Mr Ortmann's lawyer, Grant Illingworth, told the High Court that the district court had "displayed extraordinary disinterest" in the men's arguments for both the stay application and against their extradition.
Crucially, the court had not let the men present evidence of unlawful US behaviour, he said.
"[That includes] a massive search and seizure, manufacturing a situation of urgency in order to get procedural shortcuts ... covering up the unlawful activities that preceded the [arrests], downstream attempts to cover that up including a police officer giving incorrect information to this court, [and] unlawfully sending clones of hard drives overseas."
The district court failed to give any of those issues "meaningful judicial assessment" and its decision to throw out the stay application should be quashed, Mr Illingworth said.
"It was a most serious breach of the judge's duties and that alone should cause this court the deepest concern."
The only remedy was to stay the extradition attempt indefinitely, he said.
Before the appeal started this morning, Mr Dotcom's lawyer Ron Mansfield made an application to livestream the entire proceeding, under the in-court media guidelines.
"This is a case of the internet age and as such has attracted significant academic and media interest," Mr Mansfield said.
The issues being discussed could set both national and international legal precedents, he said.
"However ... the chances of such interest being satisfied by the necessarily brief footage by television or radio is negligible."
There was a risk of "unbalanced" reporting from conventional media and the livestream would provide appropriate checks and balances, Mr Mansfield said.
Mr Dotcom had hired a cameraman to film the case and it would be livestreamed on YouTube, Mr Mansfield said.
Judge Murray Gilbert was expected to hear the rest of the application tomorrow and make a ruling before the main portion of the appeal got under way.
The appeal is due to last eight weeks, though Mr Dotcom's lawyers have said they expect the case to be appealed by either side through all stages of the legal system up to the Supreme Court - a process that could take many more years to complete.
Unlike the early years of his case, when he released a widely-panned music album, funded a political party and hosted lavish parties, Mr Dotcom has kept a comparatively low profile since the extradition hearing.
In December last year, as his legal bills piled up, he moved out of the mansion he had been renting in Coatesville, north of Auckland - the site of the armed raid in which he was arrested in 2012.