  We study the spectral heat content for a class of open sets with fractal
boundaries determined by similitudes in $\mathbb{R}^{d}$, $d\geq 1$, with
respect to subordinate killed Brownian motions via $\alpha/2$-stable
subordinators and establish the asymptotic behavior of the spectral heat
content as $t\to 0$ for the full range of $\alpha\in (0,2)$. Our main theorems
show that these asymptotic behaviors depend on whether the sequence of
logarithms of the coefficients of the similitudes is arithmetic when $\alpha\in
[d-\b,2)$, where $\b$ is the interior Minkowski dimension of the boundary of
the open set. The main tools for proving the theorems are the previous results
on the spectral heat content for Brownian motions and the renewal theorem.
