Zero thermal expansion (ZTE) materials are highly desired in modern industries where high-precision processing is necessary. However, ZTE materials in pure form are extremely rare. The most widely used are Invar alloys, where the ZTE is intimately associated with spontaneous magnetic ordering, known as the magnetovolume effect (MVE). Despite tremendous studies, there is still no consensus on the microscopic origin of MVE in Invar alloys. Here, we report the discovery of room-temperature isotropic ZTE in a pure-form cobaltite perovskite, A-site disordered La0.5Ba0.5CoO3-x. The temperature window of the anomalous thermal expansion shows large tunability by simply altering the oxygen content, making this material a promising candidate for practical applications. Furthermore, we unveil with compelling experimental evidence that the ZTE originates from an isostructural transition between antiferromagnetic large-volume phase and ferromagnetic small-volume phase, which might shed light on the MVE in Invar alloys.