Question
Why do yogis prefer bathing with cold water?
Answer
Probably because hot (specially boiled) water isn't considered pure. So, bathing with/in that water will not be purifying.
See for example Manu Smriti 2.61:
अनुष्णाभिरफेनाभिरद्भिस्तीर्थेन धर्मवित् ।
शौचेप्सुः सर्वदाऽचामेदेकान्ते प्रागुदङ्मुखः ॥ ६१ ॥anuṣṇābhiraphenābhiradbhistīrthena dharmavit |
śaucepsuḥ sarvadā'cāmedekānte prāgudaṅmukhaḥ || 61 ||One who knows his duties, when desiring cleanliness, should always sip, through the proper receptacle, water which is neither hot nor frothy,—in solitude, with his face towards the North or the East.—(61)
Here it is prohibited to perform Achamana using hot water implying that hot water isn't considered pure.
The commentator Medhatithi says that here prohibition is on boiled water and not water that has naturally turned hot due to climatic condition.
Hot’ here stands for boiled; elsewhere we read that the sipping should be done ‘with unboiled water.’ So that the prohibition docs not apply to such water as is naturally hot, or has imbibed the beat of the hot atmosphere.
‘Froth’ is meant to include ‘bubbles’ also, as elsewhere we read that ‘the water should be free from froth and bubbles.’
The terms ‘through the proper receptacle’ and ‘one who knows his duties’ have been added only for the purpose of filling up the metre.
‘Desiring cleanliness’—seeking to attain cleanliness, i.e., purity. The sense is that without the sipping of water he can never be clean.
Since during winter getting hot water is not possible without heating/boiling it hence the only option is to bath in the cold water.
This does not mean that it's compulsory for everybody to do the same. I have read elsewhere that diseased or old persons (i.e. who are incapable of taking cold water baths) can bath with warm water or just sponging the body with a wet towel will also do.
And, it also does not mean that taking bath in water that is naturally hot isn't purifying. This is mentioned in one of the following verses quoted in the book "Dharma Bindu" (hosted at Kamakoti.org):
Samudra Snaana:
The general saying is:
Ashvattham manda vaaretu Saagaram parvani sprushet,
Ashvattha Saagarao sevyou kadaachana/(Ashvattha tree is worthy of touch on Saturdays and so is Samudra on Pournamis; indeed both these are worthy of worship and not to be touched indiscriminately)
Bharadwaaja Muni instructs:
Krishnaangaara chaturdashyaamyobdhou snaanam samacharet,
Tasya janma sahasreshu sarvam paapam pranashyati/
Seturnaapekshate kaalam nityam snaanam prashasyate,
Nishedhah kaalabhedasya setoranyatra karhichit/(Samudra Snaanas on Tuesdays of Krishna Paksha and Chaturdashis would destroy sins of thousands of previous births. However Setu snaanas could be done without any restrictions of days or times.)
Vyasa Maharshi describes:
Kurukshetram Gayaam Gangaam Prabhaasam Naimisham tathaa,
Tirthenyetaani sarvaani snaanakaale smaredbudhah /
Aaapayeva putaah taasaam vahnirvishodhakah,
Tasmaatsarveshu kaaleshu ushnaanbhah paavanam smritam/(All the persons of virtue should bathe at Kurukshetra, Gaya, Gangaa, Prabhasa, Naimisha and while so bathing announce the Sacred names of the Places concerned. While water purifies and is clean, Agni purifies far more and hence hot water baths are stated to be more preferable at all the places and times)
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