Thai officials believe they have a sound case and strong evidence that Aldhouse is the likely culprit in the killing of Longfellow, a former Marine.
Aldhouse's appeal appears to be based on a claim that his rights would be violated by the conditions in Thailand's jails, which do not measure up to international standards.
However, the jail in which he ended up would depend on the length of his sentence. If he was found guilty and sentenced to 15 years or less, he would serve his time in Phuket Prison.
Expat prisoners who have been inside Phuket Prison for various offences say that conditions there are crowded, but bearable.
All Thai jails are dormitory-style and not comparable to jails in the West where prisoners have individual cells or share them with a few other inmates.
The murder of Dashawn Longfellow on Phuket in August 2010 triggered enormous interest in the manhunt that followed for Aldhouse, whose Thai boxing nickname is ''Pitbull.''
He was eventually apprehended back in Britain, on arrival at London's Heathrow Airport.
Aldhouse allegedly picked a fight with Longfellow at the Freedom Bar in Rawai. When he lost the fight, he allegedly avenged his defeat by ambushing Longfellow and stabbing him to death.
Among the pieces of evidence is footage from a security camera at a nearby 7-Eleven store showing a man who looks like Aldhouse grabbing two knives, and leaving in a hurry.
The High Court decision is expected to take several weeks. If extradition is approved, it will be the first time a Briton has been extradited to Thailand to face trial.
Longfellow's family in America are strong supporters of extradition. The decision by the High Court is expected to be final and not subject to further appeal.