Has there been any research on $k$-SAT above the satisfiability threshold?
Yes, there has been. Moshe Vardi recently gave a survey talk at BIRS Theoretical Foundations of Applied SAT Solving workshop:


Moshe Vardi, Phase transitions and computational complexity, 2014.


(Moshe presents the graph of their experiment a bit after minute 14:30 in his talk linked above.)

Let $\rho$ denote the clause ratio.
As the value of $\rho$ increases beyond the threshold 
the problem becomes easier for existing SAT solvers, 
but not as easy as it was before reaching the threshold. 
There is a very steep increase in difficulty as we approach the threshold from below. 
After the threshold the problem becomes easier compared to the threshold but 
the decrease in difficulty is much less steep.

Let $T_\rho(n)$ denote the difficulty of the problem w.r.t. to $n$
(in their experiment $T_\rho(n)$ is the median running-time of GRASP on random 3SAT instances with the clause ratio $\rho$).
Moshe suggests that $T_\rho(n)$ changes as follows:


$\rho \ll$ the threshold: $T_\rho(n)$ is polynomial in $n$,
$\rho$ is near the threshold: $T_\rho(n)$ is exponential in $n$,
$\rho \gg$ the threshold: $T_\rho(n)$ remains exponential in $n$ but 
the exponent decreases as $\rho$ increases.