Given a surface $V(f) \subset \mathbb{P}^n$ for a homogeneous polynomial $f$ of degree $d$ on $\mathbb{P}^n$ and a linear transformation $g \in SL(n+1)$. Is the degree of the Hessian $H_f = V(\det (\frac{\partial f}{\partial x_i\partial x_j}))$ of $f$ equal to the degree of the Hessian $H_{f\circ g} = V(\det (\frac{\partial (f\circ g)}{\partial x_i\partial x_j}))$ of $f\circ g$? If so, why? If not, are there any restrictions under which this holds?
Many thanks in advance!