When I change the focus distance on my Pentax 55-300mm lens, the focal length appears to change. More specifically, the lens appears to zoom in as I focus towards the minimum focusing distance. This is most noticeable at the 55mm end of the lens. Is this normal?
This is called focus breathing.
With some lenses, especially macro lenses such as the Canon EF 100mm f/2.8 Macro USM, this effect can be substantial—at the minimum focusing distance, the focal length becomes about 70mm. While this is not normally an issue in still photography, it can pose a problem in video shooting as it can cause the angle of view to change while racking or following focus. Cine lenses are generally corrected for focus breathing, but lenses designed for still photography are often not corrected for this behavior.
Yes, it is normal.
Closer focusing lenses exhibit this more but it depends on the design of the particular lens. In any case, there is almost always a slight change in focal-length as focus changes. The difference will be more noticeable at the wide-end of a zoom but even prime lenses do that too.
When manufacturers specify things like focal-length and angle-of-view it is always (in my experience) with the lens focus set to infinity.