If the planet to be rejuvenated is the Earth in the far future, you could be thinking of City at World's End, a 1950 novel by Edmund Hamilton which was also the answer to this old question. What language did you read it in? The ISFDB list of translations (possibly incomplete) includes German (1952), French (1952), Portuguese (1954), Japanese (1965), French (1974), Italian (1976), and Spanish (2012). The English text is available at manybooks.net. The original magazine version (probably shorter) in Startling Stories, July 1950 is available at the Internet Archive. Here are some excerpts from the story.
The deep hole:
They had suddenly emerged into a railed gallery in a vast underground chamber. The narrow gallery was the brink of an abysmal pit—a great, circular shaft that dropped into unplumbed blackness. Kenniston stared puzzledly. He saw that big conduits led upward out of the pit, and then diverged in all directions.
"The slightly warmer air comes up from this shaft," Hubble said, nodding toward the pit. He added, "I know it sounds impossible, in our engineering experience. But I believe this shaft goes downward many, many miles. I believe it goes down to Earth's core."
"The Earth's core is incredibly hot!" Kenniston objected.
"It was hot, millions of years ago," Hubble corrected. "And as it grew cooler, as the surface grew cold, they built this domed city and maybe others like it—and sank a great shaft downward to bring up heat from the core. But Earth's core is even cooler now, almost cold. And there is only a trifle of heat from it to warm the city a little."
"So that's why they couldn't live here any more—it was the Earth heat they depended on, and that ran out," said Kenniston, a little hopelessly.
The rejuvenation plan:
"Hold on," said Kenniston. "All I know of Jon Arnol is his name. What exactly is this process? You said it was a process for the rejuvenation of cold and dying planets?"
Lal'lor answered that. "Arnol's plan is this—to start a cycle of matter-energy transformations similar to the hydrogen-helium transformation which gives a Sun its energy—to start such a nuclear cycle operating deep inside a cold planet."
Kenniston stared at him, completely stunned. "But," he said at last, "that would be equivalent to creating a giant solar furnace deep inside a planet!"
Yes. A bold, brilliant idea. It would solve the problem of the many cold and dying worlds within the Federation—since, as you know, a planet may live on its interior heat long after the parent Sun's heat has decreased."
The happy ending:
For a long, eternal moment, the dead Earth lay unstirring. Then Kenniston felt the ridge leap under his feet—once, twice, four times. The sharp grinding shocks of the capper bombs, sealing the giant shaft.
Arnol watched the quivering needles of the dials. He had ceased his trembling now. It was too late for anything, even emotion.
Deep, deep within the buried core of the Earth a trembling was born, a dilating shudder that came slowly upward to the barren rocks and touched them and was gone.
It was as though a dead beat had suddenly started to beat again. To beat strongly, exultantly, a planet reborn . . .
The pointers on the panel of dials had gone quite mad. Gradually they quivered back to normal. All but one row of them, at which Arnol and his crew stared with intensity.
Kenniston could bear the terrible silence no longer.
"Has it . . ." His voice trailed away into hoarseness.
Arnol turned very slowly toward him. He said, as though it was difficult for him to speak. "Yes. The reaction is begun. There is a great flame of warmth and life inside Earth now. It will take weeks for that warmth and life to creep up to the surface, but it will come.
