An energy-conserving lattice Boltzmann (LB) model based on the entropic theory of admissible higher-order lattice is presented in detail. The entropy supporting 'zero-one-three" lattice is used to construct a model capable of reproducing the full Fourier-Navier-Stokes equations at low Mach numbers. The proposed direct approach of constructing thermal models overcomes the shortcomings of existing models and retains one of the most important advantages of the LB methods, the exact space discretization of the advection step, thus paving the way for direct numerical simulation of thermal flows. New thermal wall boundary condition capable of handling curved geometries immersed in a multispeed lattice is proposed by extending the Tamm-Mott-Smith boundary condition. Entropic realization of the current model ensures stability of the model also for subgrid simulations. Numerical validation and thermodynamic consistency is demonstrated with classical setups such as thermal Couette flow, Rayleigh-Bénard natural convection, acoustic waves, speed of sound measurements, and shock tube simulations.