This is the Pembroke Road ventilation tower, sometimes known as the "pound tower", as it's next to the pound the Downs maintenance people use for equipment storage.

From To Keep Open and Unenclosed, the Management of the Durdham Down Since 1861, Bristol Branch of the Historical Association, 2005:

...construction of the tunnel began immediately at Cooks Folly Wood, the top of Pembroke Road and two other points. The Pembroke Road shaft was subsequently retained to ventilate the tunnel. The sinking of this shaft had reached rail level by May 1872 when Captain Beaumont's compressed air diamond drilling machine came into use. Most of the machinery associated with the diamond drilling machine was on the surface: the machine had a 4" diameter compressed air service from a 50 hp steam engine at the surface. By comparison, the engine to lift spoil and water from the tunnel was only 20 hp. The drill had to be cooled by water which was run into trucks and presumably wound to the surface. Dynamite charges were set in the holes after the dynamite had been softened by steam before use! On 10th May 1872 four pounds of dynamite exploded while being 'softened' at 2.30 a.m. blowing off the engine-house roof and smashing windows in the three houses facing the Downs at the top of Pembroke Road. Fortunately no-one was injured and when the glazier called at one house he found that the occupant had slept through the whole episode!