Is this modification of Livak's 2-delta delta CT method valid?
PCR efficiency is not usually expressed as percentage (or per-unit). Its bounds are [0 2] i.e the maximum efficiency is 2 and the minimum is 0. (Ruijter et al. 2009; also see this post for calculation of efficiency.) 

However, I just noticed that some people do express efficiency as percentage or per-unit. In that case your formula would be correct. I realize that there may be a confusion within the community; so you should know what you are calculating. For avoiding confusion lets just call this measure as amplification factor (denoted by E), which would be 1 + per-unit efficiency (as you mentioned).

When the primers have different efficiencies then you can't really use the $\Delta\Delta CT$ method. You'll end up with this formula (pretty much similar to what you have mentioned):

$$\large\text{Fold Change}=\frac{\left[E_\text{GOI}^{-CT^\text{condition}_\text{GOI}}\right]\bigg/\left[E_\text{Ref}^{-CT^\text{condition}_\text{Ref}}\right]} {\left[E_\text{GOI}^{-CT^\text{control}_\text{GOI}}\right]\bigg/\left[E_\text{Ref}^{-CT^\text{control}_\text{Ref}}\right]}=\frac{E_\text{GOI}^{-\Delta CT}}{E_\text{Ref}^{-\Delta CT}}$$

Where: 

$CT^\text{condition}_\text{GOI}$ means CT value of the gene of interest (GOI) in a given condition (other than control), and $E_\text{GOI}$ means the efficiency of the primers for your GOI.  Similarly for the reference (Ref).

See this post too.