Offenbar wird Text in spitzen Klammern nicht angezeigt. Daher der zweite Absatz noch einmal:

Als ich mir aber nun bei forvo.com die Aussprache von Theodore Roosevelt angehört habe, musste ich mit Erstaunen feststellen, dass bei dreien der vier Sprecher das t am Ende stumm ist. Die Dame, die es als einzige ausspricht, hat bisher auch keine Stimme erhalten. Ist die Aussprache mit stummen t die allgemein übliche?

bei Wikipedia nichts zur Aussprache von Franklin R., aber zu

Theodore Roosevelt (October 28, 1858 – January 6, 1919; pronounced /ˈroʊzəvɛlt/)was the 26th President of the United States.

notes
3. ^ His last name is, according to the man himself, "pronounced as if it was spelled 'Rosavelt.' That is in three syllables. The first syllable as if it was 'Rose.'" Hart, Albert B.; Herbert R. Ferleger (1989). "Theodore Roosevelt Cyclopedia" (CD-ROM). Theodore Roosevelt Association. pp. 534–535. http://www.theodoreroosevelt.org/TR%20Web%20B.... Retrieved 2007-06-10. ;
An audio recording in which Roosevelt pronounces his own last name distinctly. To listen at the correct speed, slow the recording down by 20%. Retrieved on July 12, 2007.
"How to Pronounce Theodore Roosevelt". http://inogolo.com/pronunciation/d227/Theodor.... Retrieved 2007-06-10.

The "Rosavelt" pronunciation (see #2) is the correct one. It presumably reflects the original Dutch pronunciation of the name.

All four speakers on Forvo pronounce the name this way. None of them omits the final t, although only the fourth sounds it clearly.

Are you sure you're not just hearing it because you're expecting it to be there? I've listened to it again and I cannot hear the final t except for the fourth recording.

I'm not imagining it - they all say the t. All four definitely press the tip of the tongue to the gum behind their teeth* to form a t sound, after pronouncing the l. The point is that only number 4 then sounds a very distinct t in the way that most Germans or a BBC newsreader would.

If you pronounce it "Rosavel", all four of these people would notice and think you're weird.

* There's a name for this type of sound but I can't remember what it is.