Look up the Arrhenius Equation to understand what the rate constant depends on. I believe it goes something like this:
k = pze^(-Ea/RT),
where p is collision orientation, z is collision frequency, and Ea is activation energy.
Just from this you can see why increasing activation energy decreases k and increasing temperature increases k.
The rate constant is pretty much an empirically observed term for a given reaction with its temperature and associated catalyst. As long as the temperature and catalyst are the same, changing reactant amounts won't affect its value.