Usually the the difference between your average price between $t_0$ and $T$ and the price at $t_0$ is called the Implementation Shortfall (IS).
They are a lot of references to do this, just cite these two ones:
This Figure comes from the second paper (you see how it is square-rooted):

The formula is, for a traded quantity $Q$
$$IS(Q)\simeq a \cdot \phi + b \cdot\sigma\sqrt{\frac{Q}{\rm ADV}},$$
where
- $\phi$ is the bid-ask spread
- $\sigma$ is the volatility
- $\rm ADV$ is the Average Daily Volume on the instrument.
$a$ and $b$ are two constants to be calibrated on your data. Typically: $a$ corresponds to your trading skills (if you are very smart in good liquidity chasing and if you have good execution predictors, $a$ can be close to 20%), and $b$ is somehow universal.
[EDIT] last paragraph removed (following @mbz0 remark).