Generalised Least Squares.
Mean
\begin{align*}
\widehat{\beta} &= (X_0’X_0)^{-1} X_0′ y_0 \\
&= (X’PP’X)^{-1} X’PP’ (X\beta + u)\end{align*}
But $\Omega^{-1} = PP’$
$$\widehat{\beta} = \beta + (X’\Omega^{-1}X)^{-1}X’\Omega^{-1}\underbrace{P’u}_{u_0}$$
But the transformed model satisfies the classical assumptions
$$\color{red} \Rightarrow \textsf{E}(u_0) = 0. $$
Therefore taking expectations of both sides,
\begin{align*}\color{red} \Rightarrow
\mathsf{E}(\widehat{\beta}) = \beta
\end{align*}
In general, estimator $b = \beta + Au$ for any linear estimator.
Variance
\begin{align*}
\text{var}(\widehat{\beta}) & = \mathsf{E} \bigl[ { \widehat{\beta} – \mathsf{E}(\widehat{\beta}) } { \widehat{\beta} – \mathsf{E}(\widehat{\beta}) }’ \bigr] \\
%color{red} \Rightarrow
\text{var}(\widehat{\beta}) & =\sigma^2 (X_0’X_0)^{-1}= \sigma^2 (X’\Omega^{-1}X)^{-1}
\end{align*}