Problem in deterministic time $n^p$ and not lower
For many years researchers have studied pebbling problems and emptiness/reachability problems.  Some of these problems have known unconditional resource lower bounds.

Such a problem $X$ is typically shown to have unconditional time complexity lower bounds by reducing the simulation of an $n^k$-time bounded Turing machine on a given input to an instance of $X$.  The time hierarchy theorem can then be applied to obtain the lower bound.

Here are two examples:


Pebbling games: A. Adachi, S. Iwata, T. Kasai. Some Combinatorial Game Problems Require Omega(n^k) Time. 1984
Intersecting one context-free language with k regular languages OR Intersecting k tree languages:  J. Swernofsky  and  M.  Wehar. On  the  complexity  of  intersecting  regular,  context-free, and tree languages. 2015




It's worth noting that there is a parameterized complexity class called $XP$ which contains parameterized problems that are solvable in $n^{f(k)}$ time.  A parameterized problem $X$ is $XP$-complete if every $XP$ problem is $fpt$-reducible to $X$.  There are a few known $XP$-complete problems.  Each $XP$-complete problem $X$ satisfies the property that there exist unbounded functions $f$ and $g$ such that for every $k$, $k$-$X \in DTIME(n^{f(k)})$ and $k$-$X \notin DTIME(n^{g(k)})$.

There are some parameterized problems $Y$ with finer reductions where we know that there exist $c_1 &gt; 0$ and $c_2 &gt; 0$ such that for all $k$, $k$-$Y \in DTIME(n^{c_1 k})$ and $k$-$Y \notin DTIME(n^{c_2 k})$.

Here is a post about the search for $XP$-complete problems: A list of XP-hard problems



I suspect that we could find a pebbling or reachability problem where it is solvable in cubic time, but not quadratic time (maybe intersecting three tree languages could work?).  However, this would require quite a bit of care and careful investigation of the existing Turing machine simulations.

This is a topic that I am very interested in.  Please always feel welcome to reach out if you would like to discuss this in more detail.  Thank you!