In this paper, we investigate the limiting dynamics of invariant measures of the stochastic Landau-Lifshitz-Bloch equation driven by the Stratonovich noise defined on the entire space $\R^2$. We first prove the set of all invariant measures of the stochastic equation for small noise is tight in $H^1(\R^2)$, and then prove every limit of a sequence of invariant measures of the stochastic equation must be an invariant measure of the limiting system as the noise intensity approaches zero. The main difficulty of the paper is to establish the tightness of solutions which is caused by the low regularity of solutions and the non-compactness of Sobolev embeddings on unbounded domains. To solve the problem, we first consider a family of higher-order perturbed viscous systems and then use the regularity as well as the uniform tail-ends estimates of the perturbed solutions to establish the tightness of solutions of the original equation by a limiting process.