We present two-particle $p_t$ correlations as a function of event centrality for Au+Au collisions at $\sqrt{s_{NN}}$ = 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4, and 200 GeV at the Relativistic Heavy Ion Collider using the STAR detector. These results are compared to previous measurements from CERES at the Super Proton Synchrotron and from ALICE at the Large Hadron Collider. The data are compared with UrQMD model calculations and with a model based on a Boltzmann-Langevin approach incorporating effects from thermalization. The relative dynamical correlations for Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV show a power-law dependence on the number of participant nucleons and agree with the results for Pb+Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV from ALICE. As the collision energy is lowered from $\sqrt{s_{NN}}$ = 200 to 7.7 GeV, the centrality dependence of the relative dynamical correlations departs from the power-law behavior observed at the higher collision energies. In central collisions, the relative dynamical correlations increase with collision energy up to $\sqrt{s_{NN}}$ = 200 GeV in contrast to previous measurements that showed little dependence on the collision energy.