Do methods of worship change according to yugas?


Yes, you've heard correctly. As the four Yugas progress, the world becomes more and more evil. But as a way to help people deal with all the evil that surrounds them, in later Yugas it becomes easier and easier to obtain great benefits from even small religious activity. Here is how Vyasa describes the benefits of the Kali Yuga in the Vishnu Purana:

The fruit of penance, of continence, of silent prayer, and the like, practised in the Krita age for ten years, in the Treta for one year, in the Dwápara for a month, is obtained in the Kali age in a day and night: therefore did I exclaim "Excellent, excellent, is the Kali age!" That reward which a man obtains in the Krita by abstract meditation, in the Treta by sacrifice, in the Dwápara by adoration, he receives in the Kali by merely reciting the name of Keshava.

So make sure you say my name!

Vyasa also says that it becomes easier to do your Dharma (duty):

In truth, in the Kali age duty is discharged with very little trouble by mortals, whose faults are all washed away by the water of their individual merits; by Śúdras, through diligent attendance only upon the twice-born [Brahmanas, Kshatriyas, and Vaishyas]; and by women, through the slight effort of obedience to their husbands.... [I]n the Krita and other ages great were the toils of the [twice-born] to perform their duty.

The Vishnu Puranas isn't the only scripture that describes these benefits of the Kali Yuga; here is what the sons of Rishabha say in Srimad Bhagavatam:

People in Satya-yuga are peaceful, nonenvious, friendly to every creature and steady in all situations. They worship the Supreme Personality by austere meditation and by internal and external sense control....

In Tretā-yuga, those members of human society who are fixed in religiosity and are sincerely interested in achieving the Absolute Truth worship Lord Hari, who contains within Himself all the demigods. The Lord is worshiped by the rituals of sacrifice taught in the three Vedas....

[I]n Dvāpara-yuga men who desire to know the Supreme Personality of Godhead, who is the supreme enjoyer, worship Him in the mood of honoring a great king, following the prescriptions of both the Vedas and tantras....

Those who are actually advanced in knowledge are able to appreciate the essential value of this Age of Kali. Such enlightened persons worship Kali-yuga because in this fallen age all perfection of life can easily be achieved by the performance of saṅkīrtana [chanting together].

And they even recommend two prayers to Vishnu to be said during the Kali Yuga, here and here. (Prabhupada believes these verses to be prayers to Chaitanya Mahaprabhu, who is considered an incarnation of Vishnu by Gaudiya Vaishnavas, but in my opinion they're prayers to either Vishnu himself or perhaps Venkateshwara.)

And later on in the Srimad Bhagavatam, Bhumidevi (the goddess of the Earth) herself gives a similar description of the methods of worship in the different Yugas:

The people of Satya-yuga are for the most part self-satisfied, merciful, friendly to all, peaceful, sober and tolerant. They take their pleasure from within, see all things equally and always endeavor diligently for spiritual perfection....

In the Tretā age people are devoted to ritual performances and severe austerities....

In the Dvāpara age people are interested in glory and are very noble. They devote themselves to the study of the Vedas, possess great opulence, support large families and enjoy life with vigor....

In the Kali-yuga, objects, places and even individual personalities are all polluted. The almighty Personality of Godhead, however, can remove all such contamination from the life of one who fixes the Lord within his mind. If a person hears about, glorifies, meditates upon, worships or simply offers great respect to the Supreme Lord, who is situated within the heart, the Lord will remove from his mind the contamination accumulated during many thousands of lifetimes.... Whatever result was obtained in Satya-yuga by meditating on Viṣṇu, in Tretā-yuga by performing sacrifices, and in Dvāpara-yuga by serving the Lord’s lotus feet can be obtained in Kali-yuga simply by chanting [Hari].

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