How do metal's density affects the corrosion rate?
The reason you need density is to determine the mass lost by the system, assuming that the equations you are talking about look something like:

$$
m_{loss} = k\rho S_{area} t
$$

Where $k$ is a constant defining how fast the corrosion is occurring in your system (in this case k would be in units of [length]/[time]), $\rho$ is the density of the material, $S_{area}$ is the surface area, and $t$ is time: the equation outputs the total mass of material lost. 

In more comprehensive corrosion equations $k$ would be substituted with equations describing the system causing the corrosion and may be dependent on (not a complete list): pH, temperature, surface activity, liquid flow, concentration of certain chemicals, passive layer growth, ect.