Let $h(d)$ be the class numbers of the real quadratic field $\mathbb{Q}(\sqrt{d})$. There are some heuristics of Cohen-Lenstra-Martinet about divisibility of class numbers. Do they say anything about the density of primes $p \equiv 1 \pmod{8}$ for which $h(2p)$ is a multiple of $4.$?
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The answer to the question as stated is "no". The original Cohen—Lenstra—Martinet heuristics say nothing about the $2$-Sylow subgroup of the class group of a quadratic field.
These heuristics were later modified by Gerth to take into account genus theory, see F. Gerth III, Densities for ranks of certain parts of $p$-class groups, Proc. AMS 99, and Gerth's modification does make predictions about the average behaviour of the $2$-Sylow of a "random" real quadratic number field. However, the fields you are considering form a density $0$ subfamily of the family of all real quadratic fields, so again, Gerth's heuristic says nothing about those.
However, a complete answer to the question "when is $h(2p)$ divisible by $4$" is known. This happens if and only if $p$ splits completely in $K=\mathbb{Q}(\zeta_8,\sqrt{1+i})$. One says that "$K$ is a governing field for the $4$-rank of class groups in this family". Since $K$ is a field of degree $8$, Chebotarev implies that the density of $p$ for which $h(2p)$ is divisible by $4$ is $1/8$.
For a summary of what is known about $4$- and $8$-ranks of these class groups and their governing fields, together with many references, see P. Koymans and D. Milovic, Spins of prime ideals and the negative Pell equation $x^2 − 2py^2 = −1$, Compos. Math. 155.
