We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). Thus it is used as an a posteriori error which provides information on the need to refine the mesh in the regions where discontinuities occur and to coarsen the mesh in the regions where the solution remains smooth. Nevertheless, due to the CFL stability condition the time step is restricted and leads to time consuming simulations. Therefore, we propose a local time stepping algorithm. We also use high order time extensions using Adams-Bashforth time integration technique as well as a second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: the Sod's shock tube problem, the Lax's shock tube problem and the Shu-Osher test problem.