Topology optimization is used a lot in modern industries to design products with desirable properties, like minimal compliance, weight, or stress. To achieve sufficient precision, the optimization problem as well as the underlying physical model (or state equation) often have a large number of unknowns to estimate. In classical TO procedures, the state equation has to be solved at each iteration of the optimization solver, which is equivalent to solving a very large (and different) algebraic system at each step. This computational cost is even more appaling when the procedure is combined with a linesearch to determine an almost-optimal step length, which may require additional full-order model (FOM) resolutions. On-the-fly ROMs have been previously used to reduce numerical expenses when a high-resolution simulation is needed at every iteration of the design process, without the need to sample a large number of design points in an offline phase. To further improve these methods when they are combined with a linesearch process, a new heuristic is presented. This approach is inspired by error-aware trust-region methods, which use ROMs in a classical trust-region framework to reduce the total cost of the algorithm. In this work, we propose to incorporate in the linesearch procedure an a posteriori error criterion for the POD-based ROM and restriction/relaxation coefficients for the step length, to further reduce the number of ROM updates and FOM resolutions done throughout the whole optimization process. The resulting method is a so-called error-aware backtracking linesearch (ErABLS). A brief hyperparameter study will be presented. The algorithm will be applied to classical TO benchmarks using the SIMP (Solid Isotropic Material with Penalization) material interpolation model, and to large-scale optimization problems on an HPC infrastructure.