In 1827, Richard Temple, the first Duke, felt the need to leave England due to pressing financial difficulties. He had a yacht built for the purpose, which was named after his wife, Anna Eliza. [1] 

The Anna Eliza was built at Southampton by Mr John Rubie at Southampton, who Temple angrily declared to be 'the greatest Jew and unlikable blackguard in Southampton' (HEH ST 98 Vol 2, 4 August 1827). The cost of the vessel is uncertain but a newspaper reported that it was more than £16,000:

The expense of building and fitting out the Duke of Buckingham’s yacht is said to exceed £16,000. She mounts twelve brass guns (18 and 12 pounders) with a full complement of seamen and marines, a chaplain, surgeon, &c. (Hampshire Chronicle, 13 August 1827)

After the Duke's two year tour, the Anna Eliza was put up for sale and the following advertisement was sent by Captain Wilcox to Liverpool:

For Sale. The beautiful fast Sailing Brig Yacht Anna Eliza of 254 Tons register, built at Southampton in 1827. Copper fastened and Copper sheathed, Apartments fitted in the first stile, to accommodate 30 Seamen. Ten different apartments for Gentlemen and Eight for Ships Officers or Servants, with all her equipments in which are included Ten brass Six Pounders Cannonades, now at Southampton. (HEH STG Box 446 (31); 31 March 1830)

Notes

[1] The Anna Eliza replaced his previous yacht, the Fly (which the Duke had owned from April 1814 or before; HEH ST 92, entry for 10 April 1814) :

Just as we were getting out [of Malaga] my old Yacht the Fly came in from Southampton & anchored close to us. She was sadly altered, and I should not have known her again. (HEH ST 98 Vol 2)