Perform your duty equipoised, O Arjuna, abandoning all attachment to success or failure. Such equanimity is called yoga.
Kṛṣṇa tells Arjuna that he should act in yoga. And what is that yoga? Yoga means to concentrate the mind upon the Supreme by controlling the ever-disturbing senses. And who is the Supreme? The Supreme is the Lord. And because He Himself is telling Arjuna to fight, Arjuna has nothing to do with the results of the fight. Gain or victory are Kṛṣṇa's concern; Arjuna is simply advised to act according to the dictation of Kṛṣṇa. The following of Kṛṣṇa's dictation is real yoga, and this is practiced in the process called Kṛṣṇa consciousness. By Kṛṣṇa consciousness only can one give up the sense of proprietorship. One has to become the servant of Kṛṣṇa, or the servant of the servant of Kṛṣṇa (CC-2:13:80). That is the right way to discharge duty in Kṛṣṇa consciousness, which alone can help one to act in yoga.
(Bhagavad-Gita--2:48--translation and purport).

Those whose minds are established in sameness and equanimity have already conquered the conditions of birth and death. They are flawless like Brahman, and thus they are already situated in Brahman.
Equanimity of mind, as mentioned above, is the sign of self-realization. Those who have actually attained to such a stage should be considered to have conquered material conditions, specifically birth and death. As long as one identifies with this body, he is considered a conditioned soul, but as soon as he is elevated to the stage of equanimity through realization of self, he is liberated from conditional life. In other words, he is no longer subject to take birth in the material world but can enter into the spiritual sky after his death. The Lord is flawless because He is without attraction or hatred. Similarly, when a living entity is without attraction or hatred, he also becomes flawless and eligible to enter into the spiritual sky.
(Bhagavad-Gita--5:19--translation and purport).

Samatā, equanimity, refers to freedom from attachment and aversion. To be very much attached or to be very much detached is not the best. This material world should be accepted without attachment or aversion. That which is favorable for prosecuting Kṛṣṇa consciousness should be accepted; that which is unfavorable should be rejected. That is called samatā, equanimity. A person in Kṛṣṇa consciousness has nothing to reject and nothing to accept save in terms of its usefulness in the prosecution of Kṛṣṇa consciousness.
(Bhagavad-Gita-10:4-5--purport).

Lord Caitanya accepted sannyāsa, leaving aside His householder life, to preach His mission. He has equanimity in different senses. First, He describes the confidential truth of the Personality of Godhead, and second, He satisfies everyone by knowledge and attachment to Kṛṣṇa. He is peaceful because He renounces all topics not related to the service of Kṛṣṇa. Śrīla Baladeva Vidyābhūṣaṇa has explained that the word niṣṭhā indicates His being rigidly fixed in chanting the holy name of Śrī Kṛṣṇa. Lord Caitanya subdued all disturbing opponents of devotional service, especially the monists, who are actually averse to the personal feature of the Supreme Lord.
(Sri Caitanya Caritamrta--1:3:49--purport).