Should I keep a relevant constant in the Time Complexity?
In asymptotic notation constant factors are "modded out". For example, an algorithm runs in time $O(n)$ iff it runs in time $O(2n)$ iff it runs in time $O(n/2)$.

If you want to capture such a difference meaningfully, you have (at least) two options:


Add another parameter. There could be a parameter $t$ such that your algorithm runs in time $O(tn)$. One way to think about it is that the algorithm runs in time $O_t(n)$, i.e., with a big O constant that depends on $t$, and the dependence on the big O constant on $t$ is $O(t)$. 
Count a specific operation. In some cases you might be able to count a specific type of operation more accurately. For example, perhaps one sorting algorithm performs $(2 \pm o(1)) n\log n$ comparisons on average, and another performs $(3 \pm o(1)) n\log n$ comparisons on average.