Should one research multiple independent topics while pursuing a PhD? While pursuing a PhD in an experimental field, one usually researches a given topic and conducts experiments that are rather closely related to this topic and one another. I imagine this specialization can be beneficial as one attains expertise in the research field and can work in a timely and focused manner.

However, I can also imagine that this specialization may make changing your area of research more difficult when applying for positions as a post-doc. Therefore, I am wondering about the advantages and disadvantages of researching scientific questions one is interested in, but that are irrelevant to ones PhD thesis on the side.
The main downside is that it takes a significant amount of effort to gain enough specialization in a given topic to be proficient enough to perform useful experimentation. Given that a a graduate student you're spending almost all your time familiarizing yourself with the topic you're actually performing research on, it would be very difficult to become an expert in an area unrelated to your work.

That being said, many research topics are very conducive to multi-disciplinary research. To use myself as an example, I usually tell people that I'm a cross between a neuroscientist, behavioral psychologist, signal processing engineer, and statistician. A colleague of mine started his research examining how to use the lungs as a power source for an implantable device, and ended up publishing a significant paper in the field of organic chemistry, fairly unrelated to his intended research. Becoming an expert in your own field enables you to research some pretty interesting and diverse topics.