I cannot find any information about a ship named JOSE COSTA but, as you suggest, it may have been Acosta, in which case there was a ship named SS JOSE J ACOSTA. I don't understand what you mean when you say "... next to henry t allen..." but here's what I have found.
JOSE J ACOSTA was built in 1943 by the Permanente Metals Corp., Richmond No. 1 Yard, Richmond, CA. Her keel was laid 19 November 1943, she was launched 10 December 1943, and she was delivered for service on 19 December 1943, just 30 days after keel-laying. She was sold to a private shipping firm in 1947 and scrapped in 1967. See http://shipbuildinghistory.com/history/shipyards/4emergency/wwtwo/kperm1.htm and scroll down to hull #2127. The Jose Acosta after whom the ship was named was a Jesuit missionary in Peru (1539-1600), chiefly remembered for his writings on the natural history and culture of Spanish America.
According to http://www.mariners-l.co.uk/LibShipsJon.html, JOSE J ACOSTA was sold to an Italian shipping firm in 1947 and renamed DELFIN, was sold to a Soviet company in 1963 and renamed KHIBINI, and was reportedly scrapped in 1967. My guess is that her World War II career was uneventful since I find nothing of significance with a Google search.
Additional detail about the ship may be found in a reference book entitled "The Liberty Ships," by L. A. Sawyer and W. H. Mitchell, published 1985, which may be available in public or university libraries. Most ships have at least a few sentences or a short paragraph of history, including information as to when they were lost, scrapped, or otherwise disposed of.
There is an excellent website that provides information on World War II convoys and the ships that sailed in those convoys, known as ConvoyWeb (see http://www.convoyweb.org.uk/hague/index.html). A search of ConvoyWeb finds eight convoys between April 1944 and May 1945 in which JOSE J ACOSTA sailed. You can make a "Ship Search" at the above page to find those convoys, the applicable dates, and possibly some additional detail on the individual ships in the convoys. ConvoyWeb is by no means comprehensive, meaning it will not necessarily show every convoy in which any given ship sailed.
You can make a similar search at another website, WarSailors.com (http://www.warsailors.com/search.html). I find that this website is even less comprehensive than ConvoyWeb but it is possible that either of the two websites may have specific information that the other one does not.
The website www.ancestry.com, more commonly used for genealogical research, has records of the names of passengers and crew members of ships that arrived in various U.S. ports of entry between about 1820 and 1957. A quick search of ancestry.com finds records of 307 persons (i.e., crew members) arriving in New York aboard JOSE J ACOSTA in April and October 1944 and records of another 100 persons arriving aboard the ship in San Francisco in February and May 1945. These records are presumably of both merchant marine crew and Armed Guard personnel. Ancestry.com is also not comprehensive. Ancestry.com is a subscription website; you can obtain a short-term subscription at a modest cost. The homepage of www.ancestry.com has a link to subscription options. With a subscription you will be able to search for your father's name or I can make a search for you, if you provide his name.
A search of the Armed Guard website finds two Armed Guard sailors who identified themselves as having served in JOSE J ACOSTA at some time in their careers. (These names appear on the Armed Guard website only because these men joined the Armed Guard Veterans Association at some time in the last 20 years or so. The Armed Guard website does not have information on all former Armed Guard personnel.) The men and the dates they served in JOSE J ACOSTA are as follows:
Billy M Bowers, 12/5/44 - 9/12/45
Paul P Minneti, 1/1/44 - 11/1/44
I do not know whether either of these men are still living, nor do I have contact information for them. You may wish to contact Mr. C.A. Lloyd, Chairman of the Armed Guard Veterans Association, who may have information on these individuals. Contact information for Mr. Lloyd is:
C.A. Lloyd, Chairman
U.S.N. Armed Guard World War II Veterans Association
115 Wall Creek Drive
Rolesville, NC 27571
For sources of photographs of World War II-era merchant ships, see this page: http://www.usmm.org/photosource.html.
I hope this information will be useful to you.
Ron Carlson, Webmaster
Armed Guard website www.armed-guard.com