beta-decay properties of nuclei are investigated within the relativistic nuclear energy density functional framework by varying the temperature and density, conditions relevant to the final stages of stellar evolution. Both thermal and nuclear pairing effects are taken into account in the description of nuclear properties and in the finite-temperature proton-neutron relativistic quasiparticle random-phase approximation (FT-PNRQRPA) to calculate the relevant allowed and first-forbidden transitions in the beta decay. The temperature and density effects are studied on the beta-decay half-lives at temperatures T = 0-1.5 MeV and at densities rho Y-e = 10(7) g/cm(3) and 10(9) g/cm(3). The relevant Gamow-Teller transitions are also investigated for Ti, Fe, Cd, and Sn isotopic chains at finite-temperatures. We find that the beta-decay half-lives increase with increasing density rho Y-e, whereas half-lives generally decrease with increasing temperature. It is shown that the temperature effects decrease the half-lives considerably in nuclei with longer half-lives at zero temperature, while only slight changes for nuclei with short half-lives are obtained. We also show the importance of including the de-excitation transitions in the calculation of the beta-decay half-lives at finite-temperatures. Comparing the FT-PNQRPA results with the shell-model calculations for pf -shell nuclei, a reasonable agreement is obtained for the temperature dependence of beta-decay rates. Finally, large-scale calculations of beta-decay half-lives are performed at temperatures T-9(K) = 5 and T-9(K) = 10 and densities rho Y-e = 10(7) and 10(9) g/cm(3) for even-even nuclei in the range 8 <= Z <= 82, relevant for astrophysical nucleosynthesis mechanisms.