I have heard of the earth complaining to the Gods about "bhubhara" - the weight of living things carried by her. The Gods then make tremendously destructive wars happen to reduce the earth's burden.
Is this idea found in scripture?
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Yes, the idea is found in scriptures like Puranas.
Quoting from the book "Purana Saramsha (Essence of all Puranas)":
Why Draupadi was wedded to Pancha Pandavas?
Bhu Devi requested Indra to lessen the burden of Evil Forces on Earth. Indra asked Surya, his son Yamadharma, and Vayu Deva to pass on his own virility to Devi Kunti thrice and once to Devi Madri. Simultaneously, a pious Sage awarded a boon to Kunti even when she was a Kanya (Virgin); out of curiosity Kunti desired to test the boon and invoked Surya. The latter appeared and blessed Kunti. She got conceived and had to suffer the situation by keeping the child in a floating box left it on a water body nearby. Sighting a child in a box, Radha a childless wife of a charioteer in the King’s Court adopted the child and since the latter was born with Kundalas on his ears was named as Karna or Radheya. After her wedding with King Pandu the latter was cursed by a mating deer couple which were hurt and had a curse that the moment he tried to meet his wives in bed, he would die. Kunti pacified King Pandu that she knew a Mantra by virtue of which she could invoke Devas and that was how Yudhishtar was born of Dharma Raja, Bhima was born of Vayu, Arjun of Indra himself, while Madri the younger wife of Pandu gave birth from Ashwini Kumars-but all the Devas carrying Indra’s own veerya. Draupadi was Indra’s wife Sachi Devi herself as she was born as Yagneshani emerged from Agni and Yana Purusha was Indra who divided his radiance among his own progeny of Karna and Pancha Pandavas. Hence there was no sin involved at all!
(Source: Markandeya Purana)
And, Bhudevi is Earth.
Another reference from the Devi Bhagavtam, Book 6, Chapter 11 is as follows:
1-10. Janamejaya said :-- “O King of the Brâhmanas! You said that Râma and Krisna took their incarnations to relieve the burden of earth. One great doubt arises in my mind on this point. At the end of the Dvâpara Yuga, the Earth, burdened and oppressed very much, assumed, in anguish, the form of a cow and took refuge under Brahmâ. Brahmâ, then, went with the Earth to Visnu, the Lord of Laksmî, and thus prayed, “O Bibhu! Let You, with all the other gods, incarnate soon on earth at the house of Vâsudeva to relieve the Earth of Her load, as well as to protect the righteous.”
And, the burden of earth is not the weight of all beings residing on it but specifically the burden of the evil-doers.
11-14. Vyâsa said :-- O King! As the Yuga changes, so the people changes in course of time. Nothing can alter its course, for this is caused by the Yuga Dharma (the Dharma peculiar to each Yuga). Therefore if all the subjects that are considered wicked and vicious according to the law of the Yuga Dharma, then this creation would be destroyed; hence Krisna killed only those Dânavas and vicious Ksattriyas that were really the burden of Earth.
(Devi Bhagavatam, from the same Chapter)
One of the story is in Mahabharat.
Narada said, 'Listen now, O monarch, to the following narrative of rather lengthy details as these had occurred. I myself heard it formerly, O king! Endued with great energy, the Grandsire, at the time of the creation of the universe, created a large number of living beings. These multiplied greatly, and none of them met with death. There was no part of the universe that was not overcrowded with living creatures, O thou of unfading glory! Indeed, O king, the three worlds seemed to swell with living beings, and became as it were breathless. Then, O monarch, the thought arose in the Grandsire's mind as to how he should destroy that overgrown population. Reflecting on the subject, the Self-born, however, could not decide what the means should be by which the destruction of life was to be brought about.
Lord Brahma started to kill people.
Thereupon, O king, Brahman gave way to wrath, and in consequence of his wrath a fire issued out of his body. With that fire born of his wrath, the Grandsire burnt all the quarters of the universe, O monarch. Indeed, that conflagration born of the Divine lord's anger, O king, burnt heaven and earth and the firmament and the whole universe with all its mobile and immobile beings.
In Next section Lord Brahma tells Lord Shiva about the overweight on earth.
The lord of all created beings said, 'I am not angry, nor it is my wish that all the created beings should cease to exist. It is only for lightening the burthen of the earth that destruction is desirable. The goddess Earth, afflicted with the weight of creatures, solicited me, O Mahadeva, for destroying them, especially as She seemed to sink under their burthen into the water. When after exercising my intelligence for even a long while I could not hit upon the means by which to accomplish the destruction of this overgrown population, it was then that wrath took possession of my breast.'
Lord Shiva stopped him and suggested the idea of REBIRTH.
O lord of all the lords of the universe. All this mobile and immobile universe of life, O lord of the universe, hath sprung from thee. Pacifying thee, O god of gods, I beg of thee that living creatures may repeatedly come back into the world, undergoing repeated deaths.'