The Hate Crime and Public Order (Scotland) Act 2021, external creates a new crime of "stirring up hatred" relating to age, disability, religion, sexual orientation, transgender identity or being intersex.
The maximum penalty is a prison sentence of seven years.
A person commits an offence if they communicate material, or behave in a manner, "that a reasonable person would consider to be threatening or abusive," with the intention of stirring up hatred based on the protected characteristics.
Stirring up hatred based on race, colour, nationality or ethnicity was already illegal in Great Britain under the Public Order Act 1986 but, in an attempt to streamline the criminal law in Scotland, that too is now part of the Hate Crime Act.
The bar for this offence is lower than for the other protected characteristics, as it also includes "insulting" behaviour, and as the prosecution need only prove that stirring up hatred was "likely" rather than "intended".
In England and Wales stirring up hatred over race, religion or sexual orientation by threatening behaviour remains illegal.
The hate crime law in Scotland includes more protected characteristics.
Mr Hay said his central concern was that Police Scotland "haven't been provided with any additional resources in terms of dealing with this piece of legislation."
Just last month the national force said it was no longer able to investigate every "low level" crime, including some cases of theft and criminal damage.
It has, however, pledged to investigate every hate crime complaint it receives.
BBC News understands that these will be assessed by a "dedicated team" within Police Scotland including "a number of hate crime advisers" to assist officers in determining what, if any, action to take.
At a recent meeting of the Scottish Police Authority board, Chief Constable Jo Farrell said her force would apply the act "in a measured way" under "close scrutiny."
Debate about the act has also resurrected a long-running row about how police forces throughout Britain record allegations of hate crimes where no charges are brought.
The Conservative MSP Murdo Fraser has been leading a campaign against the recording of what are termed "non-crime hate incidents" after he discovered that Police Scotland had recorded a comment he had made on social media as a hate incident, despite deciding that no crime had been committed.
More generally, the publicity about the new law, and the controversy surrounding it, has prompted warnings that it could lead to self-censorship.
Roddy Dunlop KC, dean of the Faculty of Advocates, said it could have a "chilling effect" on free speech.
And some feminist groups have raised concerns that the absence of sex as a protected characteristic in the legislation leaves women unprotected.
The Scottish government has promised to introduce a separate misogyny law "in due course" following a report by the barrister, Helena Kennedy KC.