Anthony Wood Frost was born at 5 Medley Street in Hull, Yorkshire, England on 2 January 1874 and he was later baptised on 26 January 1874 in Holy Trinity Church in that city.

He was the son of George William Frost (1846-1918), an engine fitter, and Elizabeth Wood (1845-1908), natives of Hull and Lincolnshire, respectively who had married in Yorkshire on 24 November 1868. 

He had nine known siblings: twins Charles Herbert and Jenny Wood  (b. 1869), Edmund (b. 1871), Kate (1872-1967, later Mrs Charles Dunlop),  Harold (1877-1964), Walter Stanley (1878-1944), Aaron (b. 27 September 1880), Septimus (b. 9 November 1883) and Annie (31 January 1886-1966, later Mrs Stephen Reginald Bossence).

He first appears on the 1881 census as a visitor to the home of his maternal aunt Harriet Frost at 1 Medley Street, Sculcoates, Yorkshire. His family moved to Belfast around this time where his father worked for Harland & Wolff. By 1886, at the time of youngest sister Annie's birth, their residence was 87 Woodstock Road in east Belfast.

Anthony appears on the 1901 Irish census living at 12 Mount Street off the Ormeau Road in south Belfast, Co Down and he was described, like his father, as an engine fitter.

He was married in Belfast on 12 June 1901 to former bookkeeper Elizabeth "Lizzie" Jane Hamilton (b. 15 June 1874 in Ballyclare, Co Antrim), the daughter of John Hamilton, a shoemaker and shopkeeper, and Susan, née Abernethy. Their respective Belfast addresses were listed as 12 Mount Street and 117 Albert Bridge Road and Frost was described as an engineer. 

The couple had four children: Robert John Hamilton (b. 29 July 1902-1986), Walter Stanley (b. 5 November 1905-1985), Marjorie Wood (b. 29 October 1907-1995, later Mrs Harry McCormick) and Doreen Abernethy (b. 7 June 1910-1973, later Mrs Samuel Hamilton Dunlop). The family appear on the 1911 census of Ireland living at 11 Sunbury Avenue, Pottinger, east Belfast and Anthony was described as a foreman engine fitter. His mother died on 6 May 1908 and his father on 19 December 1918; both are buried in Dundonald Cemetery, Co Down.

In the service of Harland & Wolff for twenty-four years, Frost commenced work at Queen's Island as a machine boy on 4 January 1888 before being appointed apprentice fitter the following year. After completion of his apprenticeship in March 1894 he spent two years at sea to gain practical experience.

Following Frost's return to the shipyard he was promoted to leading hand in March 1901. When his father retired in 1907 young Frost succeeded him as outside foreman engineer. He had been in charge of the overhauls of the machinery aboard HMS Endymion and Hermes and during the trials of HMS Enchantress, Hibernia and Minotaur. He went on to supervise the installation of the plant aboard Olympic and Titanic. Described as one of the most energetic and capable foremen at Harland & Wolff and perhaps on the eve of a promotion, Frost was one of the nine-strong "guarantee group" of Harland and Wolff employees chosen to watch over the smooth running of the Titanic's maiden voyage. 

Anthony Frost, like the rest of his counterparts, died in the sinking; his body, if recovered, was never identified.

Frost's estate, valued at £254, was administered to his widow on 12 July 1912.

His sister Annie was married only weeks after the disaster on 6 July 1912 to engine fitter Stephen Reginald Bossence of Woodstock Road, Belfast. She died on 23 August 1966 and was buried in Roselawn Cemetery, Belfast. Their mother Elizabeth had died on 6 May 1908 and their father on 19 December 1918; both are buried in Dundonald Cemetery, Co Down.

Anthony's widow Lizzie was never remarried and she remained in Belfast for the rest of her life. She later lived at 2 Hillcrest Gardens and died as a result of senility and heart disease on 23 February 1961 and was buried in Roselawn Cemetery, Belfast. Her last surviving child Marjorie died in Belfast on 17 July 1995 aged 87. 

Anthony's last known surviving sibling Kate Dunlop died in Belfast on 16 June 1967 aged 95.

Anthony Wood Frost is remembered on his wife's headstone in Roselawn Cemetery Belfast, Ireland (section E, plot 555).