Units of the rate constant for reactions whose order is greater than 1
As an example if the reaction is $\ce{A + B -&gt; C}$ the rate expression for the appearance of $\ce{C}$ is

$$\frac{\mathrm d[\ce{C}]}{\mathrm dt} = +k[\ce{A}][\ce{B}].$$

The units have to be the same on both sides of this equation, these are concentration/time on the left and to be the same on the right the rate constant $k$ has to have units $(1/\text{concentration})(1/\text{time})$ usually expressed as $\mathrm{dm^3\,mol^{-1}\,s^{-1}}.$ 

It is true that the numerical value of the rate constant usually depends of several things such as temperature, but the units remain the same for each type of rate constant, 0th,  1st, 2nd, 3rd simply because the rate equation units have to balance.