This paper is a contribution to semiclassical analysis for abstract Schrödinger type operators on locally compact spaces: Let $X$ be a metrizable seperable locally compact space, let $μ$ be a Radon measure on $X$ with a full support. Let $(t,x,y)\mapsto p(t,x,y)$ be a strictly positive pointwise consistent $μ$-heat kernel, and assume that the generator $H_p\geq 0$ of the corresponding self-adjoint contraction semigroup in $L^2(X,μ)$ induces a regular Dirichlet form. Then, given a function $Ψ: (0,1)\to (0,\infty)$ such that the limit $\lim_{t\to 0+}p(t,x,x)Ψ(t)$ exists for all $x\in X$, we prove that for every potential $w:X\to \mathbb{R}$ one has $$ \lim_{t \to 0+} Ψ(t)\mathrm{tr}\big(\mathrm{e}^{ -t H_p + w}\big)= \int \mathrm{e}^{-w(x) }\lim_{t \to 0+}p(t,x,x) Ψ(t) dμ(x)<\infty $$ for the Schrödinger type operator $H_p + w$, provided $w$ satisfies very mild conditions at $\infty$, that are essentially only made to guarantee that the sum of quadratic forms $ H_p + w/t$ is self-adjoint and bounded from below for small $t$, and to guarantee that $$ \int \mathrm{e}^{-w(x) }\lim_{t\to 0+}p(t,x,x) Ψ(t) dμ(x)<\infty. $$ The proof is probabilistic and relies on a principle of not feeling the boundary for $p(t,x,x)$. In particular, this result implies a new semiclassical limit result for partition functions valid on arbitrary connected geodesically complete Riemannian manifolds, and one also recovers a previously established semiclassical limit result for possibly locally infinite connected weighted graphs.