The title of Baron Burghersh has been created three times in the Peerage of England.
It was first created by writ for Robert de Burghersh on 12 November 1303. Robert had three sons, the eldest of which, named Stephen, became the second Baron. According to modern peerage law, the title would have descended to Maud, Stephen's only daughter, then to her son Sir Walter Paveley, and afterwards to his son, also named Walter. However, there is no evidence that Maud or her descendants ever used the title. After Walter's death the title presumably became abeyant or extinct.
The title was created again by writ for Robert's third son Bartholomew de Burghersh on 25 January 1330. It descended through the families of Despencer and Beauchamp, before becoming abeyant in 1449.
The title was created again on 29 December 1624 for Francis Fane. Francis was made Earl of Westmorland at the same time, and both titles are currently held by Anthony Fane, his descendant.
After Anne's death at the age of five, the Barony of Burghersh fell into abeyance between her aunts. (Her Earldom passed to her aunt Anne Beauchamp, her father Henry's only full sister; her husband Richard Neville then became jure uxoris 16th Earl of Warwick.)
Cokayne, George E.; Vicary Gibbs, Peter; Doubleday, Harry; Howard de Walden, Lord David (1958). The Complete Peerage of England, Scotland, Ireland and Great Britain extant, abeyant, dormant and extinct. Vol. XIV vols.