What does thermal stability of compounds depend on?
Lattice energy might be important if the decomposition took place in the solid phase. 
But Li2CO3 melts at 723 C and boils (with decomposition) at 1310 C. Na2CO3 melts at 851 C and decomposes before boiling, according to an older CRC Handbook and Wikipedia, which seems to be pretty current.  According to those numbers, thermal stability of sodium and lithium carbonates seems fairly similar, and decomposition does not take place because of solid issues. 

K2CO3 melts at 891 C, decomposes before boiling. Rb2CO3 melts at 837 C, decomposes at 900 C. Cs2CO3 melts/decomposes at 610 C. Lattice energy might be important with Rb and Cs, but thermal stability seems to become less stable as you go down the group, not more so. 

The first three seem fairly stable for several degrees above the melting point. We could consider the melt to be a mix of CO3-- anions with the metal cations, all oxygen-bonded (similar to hydrogen bonding!), but much looser than in a crystal. The thermal stability of the CO3-- anion does not seem to be very dependent on the ionic size of these three metals (Li+, Na+, K+).

So, I would ask the textbook author to substantiate his claims.