Consider a smooth conic bundle $X\rightarrow \mathbb{P}^1$ with discriminant of degree $d$ (the locus of $\mathbb{P}^1$ over which the fibers are reducible conics). There is a formula for $(-K_X)^2$ that reads $(-K_X)^2 = 8-d$. So that the volume depends only on the degree of the discriminant.
Now, consider conic bundles $Y\rightarrow\mathbb{P}^2$. For instance, if $Y\subset\mathbb{P}^2\times\mathbb{P}^2$ is a divisor of bidegree $(4,2)$ I got $(-K_Y)^3 = -6$. Note that in this case $d = 12$.
If $Y\subset\mathbb{P}(\mathcal{O}_{\mathbb{P}^2}(2)\oplus \mathcal{O}_{\mathbb{P}^2}(1)\oplus \mathcal{O}_{\mathbb{P}^2})$ given by an equation of the form $$ a_0x^2+a_1xy+a_2xz+a_3y^2+a_4yz+a_5z^2 = 0 $$ where the $a_i$ are polynomials on the base $\mathbb{P}^2$ of degree respectively $6,5,4,4,3,2$. Note that also in this case $d = 12$.
So it seems that unlike the case of surfaces in higher dimension the is no formula for $(-K_Y)^n$ where $n$ is the dimension of the conic bundle $Y$.
Is this correct and if so is there a conceptual explanation for this fact?