What would be some security implications of a fixed pathname (although probably in a system directory when it would be deployed as a system service) rather than a relative (out of current directory)? If the answer is "it depends", how so?
In general: an absolute path name is more clear than a relative one. It is not open to interpretation based on the context like a relative path name is. The actual meaning of a relative name depends on the current directory, on environment variables (like PATH) etc, while an absolute path name most times means the same thing.
Therefore an absolute path name is almost always the more secure choice. Of course one is probably able to deliberately construct some scenario where the absolute path is the more insecure choice and likely there will be even such scenarios out there in the wild - developers are often creative in inventing their own approaches of how things should work and of ignoring established ones.
There is one scenario I could think of though, where a relative name can be less ambivalent: if chroot or similar things are used, then for example a symlink using an absolute path can result in a different target file inside and outside the chroot, while a symlink using a relative path might still point to the same file. And this difference might actually result in a security issue.