Building on Daniel Ballinger's answer to 'How to get recent items', the difference is really minor:
/services/data/v26.0/recent
gives you recent items like in the sidebar. "Object-specific" are well - object specific:
/services/data/v26.0/sobjects/Account
/services/data/v26.0/sobjects/Opportunity
etc.

Or raw output:
"recentItems" : [ {
"attributes" : {
"type" : "Opportunity",
"url" : "/services/data/v26.0/sobjects/Opportunity/0067000000AH3M7AAL"
},
"Id" : "0067000000AH3M7AAL",
"Name" : "University of AZ Portable Generators111"
}, {
"attributes" : {
"type" : "Opportunity",
"url" : "/services/data/v26.0/sobjects/Opportunity/0067000000AH3M1AAL"
},
"Id" : "0067000000AH3M1AAL",
"Name" : "Grand Hotels Kitchen Generator"
}, {
"attributes" : {
"type" : "Opportunity",
"url" : "/services/data/v26.0/sobjects/Opportunity/0067000000AH3MEAA1"
}...