Stochastic thermodynamics in mesoscopic chemical oscillation systems is discussed on the basis of chemical Langevin equation for the state variables with particular attention paid to a parameter region close to the deterministic Hopf bifurcation. The Langevin dynamics defines stochastic trajectories in the state space and therefore trajectory dependent entropy and entropy production according to the schemes proposed by Udo Seifert (Phys. Rev. Lett. 2005, 95, 040602). The total entropy change along a stochastic trajectory obeys the fluctuation theorems. By using the stochastic normal form theory, we derive explicit theoretical expressions for the mean entropy production in the state. The resulting entropy production in the large system volume V limit can scale linearly or independent with V when the control parameter is above or below the Hopf bifurcation while it is of V-1/2 at the bifurcation. We verify the above relations by direct simulation with a stochastic circadian clock model.