In general, finding least-cost paths with arbitrary time-dependent edge costs is NP-hard, assuming that either (a) waiting at nodes is forbidden and your costs are not FIFO-preserving (e.g., see here for brief discussion on this complexity, as well as others: https://www.cs.ucsb.edu/~suri/psdir/soda11.pdf) or (b) the cost you are trying to optimize is not simply pure travel time itself (e.g., see http://algo2.iti.kit.edu/download/gen_obj_func_tch.pdf).
As an illustrative example of this latter situation (which seems most similar to yours) and continuing with your suggested notation, it is shown in the second paper above that it is NP-hard for the case in which the time-dependent edge cost is defined in the form $c_e(t) = f_e(t) + d_e$, where $f_e$ is the (typical) time-dependent travel-time function that returns the travel time along the edge for a particular time of day $t$ and $d_e$ is some arbitrary time-neutral cost function. It is not hard to show that similar hardness results for other such cases as well, where, e.g., the function uses a multiplier instead of addition (i.e., $c_e(t) = f_e(t) \cdot d_e$), and so on.
Regardless, one can derive pseudo-polynomial time bounds on such cases if, e.g., you are representing your time instants as integer values and assuming there are only a finite number of possible time instants bounded by $T$. In this way, you only need to maintain a single least-cost path per time instant $t$ for any given node in the graph, which would give you the running time of $O(T(m + nlogn))$, assuming non-negative cost functions. Taking this idea a step further, since your particular use case assumes that "each edge takes one unit of time to traverse", then this automatically gives you a strongly-polynomial-time algorithm since, in your case, $T \in O(n)$ (because each such path contains $\leq n-1$ edges).