  We analyze the thermodynamic properties of a simplified model for folded RNA
molecules recently studied by G. Vernizzi, H. Orland, A. Zee (in {\it Phys.
Rev. Lett.} {\bf 94} (2005) 168103). 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 $N \times N$ 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 $\rm{Mg}^{++}$ in solution.
