We analyze the thermodynamic properties of a simplified model for folded RNA molecules recently studied by Vernizzi, Orland, and Zee [Phys. Rev. Lett. 94, 168103 (2005)]. The model consists of a chain of one-flavor base molecules with a flexible backbone and all possible pairing interactions equally allowed. The spatial pseudoknot structure of the model can be efficiently studied by introducing a N×N Hermitian random matrix model at each chain site, and associating Feynman diagrams of these models to spatial configurations of the molecules. We obtain an exact expression for the topological expansion of the partition function of the system. We calculate exact and asymptotic expressions for the free energy, specific heat, entanglement, and chemical potential and study their behavior as a function of temperature. Our results are consistent with the interpretation of 1/N as being a measure of the concentration of Mg²⁺ in solution.