This article is from the Photographic Lenses FAQ, by David Jacobson with numerous contributions by others.
Assuming that you frame the subject the same way, and that the object (subject) distance is small compared with the hyperfocal distance for the shortest focal length being considered, the front and rear depths of field are approximately equal and constant regardless of focal length. (See Question 14.)
However, there are two situations in which focal length does matter.
First, when the focal length is short enough, the hyperfocal distance, which varies with the square of the focal length, will not be many times longer than the object distance, violating the condition above. In this case the front depth of field is smaller and the rear depth of field is larger, with the latter extending to infinty about when the hyperfocal distance is less than the object distance.
Second, the focal length of the lens also has a big effect on how fuzzy very distant points appear. Specifically, if the lens is focused on some nearby object rendered with magnification M, a point at infinity will be rendered as a circle of diameter c, given by
c = f M / N
which shows that the distant background point will be fuzzed out in direct proportion to the focal length. (See the lens tutorial for some graphs that may make this more intuitive.)