A court sentenced Myanmar migrant workers Zaw Lin and Win Zaw Htun, both 22, to death on December 24 after finding them guilty of murdering tourists Hannah Witheridge, 23, and David Miller, 24, on a beach on the island of Koh Tao in 2014.
Witheridge had been raped and bludgeoned to death, according to police, and Miller suffered severe head injuries and was left to drown in the surf.
But the investigation drew allegations of police incompetence, torture and mishandling of evidence.
Police rejected the complaints but protests erupted in neighbouring Myanmar after the verdict, with many people there believing the two workers, who denied killing the tourists, were scapegoats.
Some rights workers say they believe police forced the two into confessing under duress. Police denied that.
In a message posted on the 'We Are Anonymous' Facebook page the hacking group said it had shut down Thai judicial websites in protest over the verdict.
"Anonymous shuts down all Thai Court of Justice website in protest over the Koh Tao murder verdict. Anonymous is supporting the campaign to ask tourists to boycott Thailand," the group said.
It also posted a black and white graphic of a man wearing a Guy Fawkes mask, an image associated with the Anonymous movement, with a blindfold over his eyes and the words: "Failed Law. We Want Justice!"
The group listed more than 200 websites it said it shut down.
The Civil Court of Thailand's main website was not immediately accessible but the Courts of Justice website, which the group said it had hacked, was running as normal late on Wednesday.
Police Colonel Somporn Daengdee, deputy chief of the police Technology Crime Suppression Division, said police had not received reports of government websites being hacked but were looking into it.
In an emotionally charged Facebook post on Tuesday, Witheridge's sister criticised the investigation and the police.
Miller's family has said it supported the court verdict and the work of the police.
This month, Anonymous, a loose-knit international network of activist hackers, said it was responsible for cyber-attacks on Thai police websites in protest against the Koh Tao verdict.
Police confirmed the attacks but said they were not on confidential data. Reuters was unable to verify who carried out the attacks.
© Thomson Reuters 2016
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