Myanmar migrant workers Zaw Lin and Wai Phyo were found guilty in late December of the rape and murder of Hannah Witheridge and the murder of David Miller on the resort island of Koh Tao in September2014.
The death sentences handed out by a Thai court prompted a series of protests in neighbouring Myanmar, with protesters saying that the workers were scapegoats, and that the trial was flawed.
"Apparently these attacks are in response to the recent [Koh Taomurder] case," said Police General Suppaset Chokchai.
"We are currently verifying whether these claims are true and how best to proceed."
Searches for the Thai Metropolitan Police website on Google returned with the result: "Blink Hacker Group, Failed Law We want Justice!"
This and other related sites appeared to be back in operation by 4:40pm (1140 GMT).
On Monday, the Thai embassy in Yangon said that consular services would remain closed until Friday "due to the ongoing demonstration which have caused difficulties in access to the compound."
Myanmar's ambassador to Thailand, U Win Muang, has told reporters that the embassy was monitoring the appeals process closely and hoped that the workers would be treated fairly.
DPA