Fixed primes in discrete logarithm The discrete logarithm problem is to find $z$ when the inputs are $g,h,p$ where $g^z\bmod p$.

Supposing if you fix $p$ then does the problem become any easier to attacks and is there a $(\log p)^c$ complexity algorithm?
The best algorithm for computing discrete logarithms are based on families and variations of methods known as "number field sieve" and "function field sieve". They all share a common structure. In particular, they are divided in several phases, only the last of which depends on the value whose discrete logarithm must be computed: all the previous steps require only the order of the field.

I did some research. I do not know about the computational complexity of each step, this would involve some more research but can probably be found in some of the articles and surveys of Antoine Joux. However, I'll focus on the case which seems to be your target: discrete logarithm in prime order field. For that case, the actual record is a discrete log in a field of order $p$, with $bitsize(p) = 530$. You'll find Here the report on this record. The computation is divided in five phases: polynomial selection, sieving, filtering, linear algebra, and actual computation of the discrete logarithm. Among them, the sieving is by far the dominant cost: the report indicates it took approximately 50 core-years on a 2-GHz Intel E5-2650. This phase depends only on the order $p$ of the field. The report mention that once the four first phases have been performed, "Computing one individual logarithm required a few hours". Therefore, Compared to the overall time, computing discrete logarithms is extremely fast once the precomputation has been made (with the order of the field only).

Although I do not know the theoretical complexity of this last step, I'm pretty sure that this is still not polynomial time (id est, $O(\log^c p)$ for some $c$), even if it is only a negligible cost in the total computation.