Sahih Muslim 3248—Ibn Uraij reported: 'Ati' reported that Jabir b. Abdullah came to perform 'Umra, and we came to his abode, and the people asked him about different things, and then they made a mention of temporary marriage, whereupon he said: Yes, we had been benefiting ourselves by this temporary marriage during the lifetime of the Holy Prophet and during the time of Abu Bakr and ‘Umar.
Some hadith claim that it was Umar, rather than Muhammad, who outlawed Mut'ah:
Sahih Muslim 3250—Abu Nadra reported: While I was in the company of Jabir b. Abdullah, a person came to him and said that Ibn 'Abbas and Ibn Zubair differed on the two types of Mut'a (Tamattu’ of Hajj 1846 and Tamattu’ with women), whereupon Jabir said: We used to do these two during the lifetime of Allah's Messenger. Umar then forbade us to do them, and so we did not revert to them.
According to Sahih al-Bukhari (Sunni Islam's most trusted collection of ahadith), Mut'ah is justified by the Qur'an itself! Consider the following verse from the Qur'an:
Qur'an 5:87—O ye who believe! Forbid not the good things which Allah hath made lawful for you, and transgress not, Lo! Allah loveth not transgressors.
Notice how this verse was used by Muhammad:
Sahih al-Bukhari 5079—We used to participate in the holy battles led by Allah's Messenger and we had nothing (no wives) with us. So we said, "Shall we get ourselves castrated?" He forbade us that and then allowed us to marry a woman temporarily by giving her even a garment and then he recited to us: "O you who believe! Make not unlawful the good things which Allah has made lawful for you." (5.87)
Hence, Muslim men who want to hire prostitutes can simply point out that both the Qur'an (the Word of Allah) and Muhammad (Islam's final prophet) allowed prostitution, and that later Muslims (like Umar) cannot overrule Allah and Muhammad.
For some of the practical consequences of Muhammad's teachings about Mut'ah, see the following articles:
"Syrian Mothers Selling Daughters as Prostitutes to Wealthy Saudis in Accordance with Sharia"
"Muta (Islamic Prostitution) Still Common in Poor Areas of Egypt"
"Hezbollah Using Prostitution (Muta) for Recruitment"
"Temporary Marriage in Iran"
For a more thorough discussion of Mut'ah in the Islamic sources, see the following article by Sam Shamoun:
"Revisiting Muhammad’s Permitting a Form of Prostitution known As Muta"