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The correlation between the mean transverse momentum, [p T], and the squared anisotropic flow, vn2, on an event-by-event basis has been suggested to be influenced by the initial conditions in heavy-ion collisions. We present measurements of the variances and covariance of [p T] and vn2, along with their dimensionless ratio, for Au+Au collisions at various beam energies: sNN = 14.6, 19.6, 27, 54.4, and 200 GeV. Our measurements reveal a distinct energy-dependent behavior in the variances and covariances. In addition, the dimensionless ratio displays a similar behavior across different beam energies. We compare our measurements with hydrodynamic models and similar measurements from Pb+Pb collisions at the Large Hadron Collider (LHC). These findings provide valuable insights into the beam energy dependence of the specific shear viscosity (η / s) and initial-state effects, allowing for differentiating between different initial-state models. © 2026 The Authors.