Study of the transverse structure of the proton is one of the major physics goals of the upcoming electron ion collider (EIC). The gluon transverse momentum dependent distributions (TMDs) form an essential focus of this effort and are important towards understanding the angular momentum contribution to proton spin as well as QCD factorization. However, very limited experimental constraints on the gluon TMDs exist currently. As the heavy quark production in lepton-nucleon DIS gets a dominant contribution from the photon-gluon-fusion process, heavy quark production makes an attractive tool to probe gluon distributions in nucleons. In this paper we present a study of heavy flavor hadron pair reconstruction at a future EIC detector with MAPS based inner tracking and vertexing subsystems to constrain gluon TMDs. We utilize the excellent track pointing resolution provided by the detector to exclusively reconstruct heavy flavor hadron pairs via their hadronic decay channels and also to develop a heavy flavor hadron tagging algorithm. Statistical uncertainty projections on azimuthal asymmetries corresponding to gluon TMDs at the EIC is evaluated. The heavy flavor tagging is found to substantially enhance the purity of heavy flavor hadron pair selection, and the statistical precision of the measurement compared to that from exclusive reconstruction. The correlation between the azimuthal angle of the transverse momentum of the gluon initiating the process and that of the corresponding heavy flavor hadron pair was also studied and found to be well correlated. This study opens up heavy flavor hadron pair measurements as an attractive channel to access gluon TMDs at the EIC.
