Can a PhD student publish a paper outside PhD topic without supervisor approval only by its own? I am a theoretical PhD student. Is it possible to submit and to publish papers in journals without supervisor approval? I am asking also from legal point of view, as well. I have talked to many people in my university and some of them don't have  a good opinion regarding single author papers since such a single author paper might have sufficient errors in it. However, the subject is not studied by anyone in the university.
There is no general reason why this should not be possible/allowed.

You will have to check the regulations of your department/university for PhD students: They may require you to seek approval of your advisor prior to submission of papers. Potentially a journal could have a similar rule in place, but I have never encountered this.

That said, I do not believe this to be advisable: If your relationship with your PhD advisor is so bad that do you not feel you can ask them about your intention to submit this paper, you should probably seek a new advisor. If you are so close to graduating that your prefer sitting it out, then rocking the boat is probably a bad idea.