In this paper, the detailed dynamic characteristics of the subgrid scale (SGS) stress tensor and heat flux are investigated through Taylor series expansion in numerical simulations of compressible isotropic turbulence. A new approximate second-order closure (ASOC) model is introduced based on the transport equations of the first-order Taylor series approximation of SGS stress tensor and heat flux. The proposed model is implemented in large eddy simulation (LES) of compressible isotropic turbulence. Detailed comparisons with direct numerical simulation (DNS) dataset using both a priori and a posteriori approaches are carried out. A priori tests show that, SGS stress tensor and heat flux have high correlations with the first-order Taylor series approximation. Their root mean square (rms) values are close to those of the first-order Taylor series approximation. In a posteriori tests, the proposed ASOC model yields good agreement with DNS dataset. Compared with the results of the dynamic Smagorinsky model (DSM) and dynamic mixed model (DMM), the ASOC model predicts better energy spectra at high wavenumbers. The probability density function (PDF) and the structure functions of velocity and thermodynamic variables are further studied, demonstrating that the statistical properties of the simulated flows are improved by the ASOC model. The numerical results illustrate the ability of the model to improve the statistical properties of the simulated flows in the context of LES. Finally, a simplified ASOC model can be derived by neglecting the effect of density gradient for low turbulent Mach number turbulence.