An energy eigenvalue expression for a sextic anharmonic interaction potential is derived by solving the Schrödinger equation analytically within the framework of the Nikiforov–Uvarov (NU) method. The resultant expression is then used to calculate the mass spectra of some heavy and light mesons, viz., c$\bar{c}$, b$\bar{b}$ , b$\bar{c}$, c$\bar{s}$, b$\bar{s}$ and b$\bar{q}$, and the results are found to be in good agreement with the other theoretical and experimental studies. The energy eigenvalue expression is further used to compute some important thermodynamic quantities like partition function, specific heat capacity, free energy, mean energy, entropy and magnetisation.