Suppose $f$ is a self-concordant(see 9.6.2) barrier of a proper cone $K$ ( solid,convex, closed and pointed) in $\mathbb{R}^n$. It looks like the value of all $\nabla f$ is just the interior of the polar dual $-K^*$.
Examples such as non-negative octant and $-\sum \log(x_i)$ and the positive semi-definite cone $\mathbb{S}_+^n$ and $-\log \det(X)$ seems to suggest it is the case.
I wonder is it true in general or is there any existing material proves this.
This is absolutely the case, and it is straight out of the seminal text, Interior Point Polynomial Algorithms in Convex Programming by Yurii Nesterov and Arkadii Nemirovskii. See Section 2.4.3, "Legendre transformation of a self-concordant logarithmically homogeneous barrier." (Or don't. The book is rather... uh... thick. :-)) I don't think this exact observation is cited there, but because of the way the gradient is involved in the Legendre transformation, this property drops out rather simply.