Not the name of a tavern, but an un-named tavern that had a grotto of some kind.
"Kensington, like other villages along main roads out of London, long attracted hostelries and drinking houses for travellers; Pepys, for one, confides to his diaries of the 1660s the record of several expeditions to Kensington and bibulous evenings in one or more taverns there (one with a grotto, but all unnamed).2. The Diary of Samuel Pepys, edd. Robert Latham and William Matthews, vol. 5, 1971, pp. 178, 180–1; vol. 9,1976, pp. 166, 170, 203."
Chart showing the number of references in each month of the diary’s entries.