Should one research multiple independent topics while pursuing a PhD?
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.