Scientists believe they could potentially detect signs of life around white dwarf stars in the near future.
White dwarf stars come to be when a star about the size of our sun burns through its store of hydrogen fuel in its core. The star's outer atmosphere expands away, leaving behind a white hot core about the size of Earth composed mostly of carbon. White dwarfs are not as hot as the stars that birthed them, but they can burn for billions of years.
Two facts about white dwarfs make them attractive targets for ET hunters. First, they can harbor planetary systems just like a normal star. Second, they are dimmer than other stars. Planets that orbit white dwarfs either formed from the debris remaining from the death of the star, or they were worlds that orbited the original star further out before spiraling inward. The dimmer light from white dwarfs would make it easier for astronomers to sight hypothetical planets with their telescopes because it would not be lost in the bright glare. Also, an Earth-like planet would be similar in size to its host star, and its transit would block a substantial amount of the star's light.
Astronomers can study the atmospheres of transiting planets by studying light that passes through them. If they detect a substantial amount of oxygen and water vapor in the atmosphere, that would be decent evidence of Earth-like life.
The James Webb Space Telescope, which will be launched by the end of the decade, will be powerful enough to detect oxygen on potential Earth-like planets orbiting white dwarfs, should they be found.