This article reports evidence for superconductivity above 40 K in the La-Ba-Cu-O (LBCO) compound system under hydrostatic pressure. The study describes the synthesis of LBCO compounds via a solid-state reaction method and examines their superconducting properties. The results show that the superconducting transition temperature (Tc0) increases with pressure, reaching 40.2 K at 13 kbar. The superconducting transition is observed as a sharp drop in resistivity (ρ) and is associated with nonbulk superconductivity, consistent with percolative superconductivity. The ρ drop is broadened and shifted to higher temperatures under pressure, but the overall Tc0 increases significantly. The study also finds that the ac magnetic susceptibility (χ) measurements indicate a diamagnetic shift, consistent with superconductivity. However, the small χ shift in the samples suggests that the superconductivity may be associated with a yet unidentified minor phase or interfaces in the LBCO compounds. The large positive pressure effect on Tc0 (~0.9×10−3 kbar−1) is unprecedented and is attributed to interfacial superconductivity due to mixed phases or concentration fluctuations. The study concludes that superconductivity above 40 K is achievable in LBCO under pressure and with optimized sample conditions. The results suggest that superconductivity at temperatures greatly exceeding 40 K is achievable in LBCO and related systems through fine tuning of the sample parameters. The study is supported by grants from the National Science Foundation and the U.S. National Aeronautics and Space Administration.This article reports evidence for superconductivity above 40 K in the La-Ba-Cu-O (LBCO) compound system under hydrostatic pressure. The study describes the synthesis of LBCO compounds via a solid-state reaction method and examines their superconducting properties. The results show that the superconducting transition temperature (Tc0) increases with pressure, reaching 40.2 K at 13 kbar. The superconducting transition is observed as a sharp drop in resistivity (ρ) and is associated with nonbulk superconductivity, consistent with percolative superconductivity. The ρ drop is broadened and shifted to higher temperatures under pressure, but the overall Tc0 increases significantly. The study also finds that the ac magnetic susceptibility (χ) measurements indicate a diamagnetic shift, consistent with superconductivity. However, the small χ shift in the samples suggests that the superconductivity may be associated with a yet unidentified minor phase or interfaces in the LBCO compounds. The large positive pressure effect on Tc0 (~0.9×10−3 kbar−1) is unprecedented and is attributed to interfacial superconductivity due to mixed phases or concentration fluctuations. The study concludes that superconductivity above 40 K is achievable in LBCO under pressure and with optimized sample conditions. The results suggest that superconductivity at temperatures greatly exceeding 40 K is achievable in LBCO and related systems through fine tuning of the sample parameters. The study is supported by grants from the National Science Foundation and the U.S. National Aeronautics and Space Administration.