Greetings.
There were different things written that are linked, but not the same.
TOM1953 explains correctly both.
Tensile strenght is affected by the samples' thickeness. For most of the properties (if not all) there are standard tests. For this case in particular, check ISO 37

2005). If you like to test this, just place a thicker sample for this test (if you can) and you will see this result.
A thick piece, is always in trouble for the reasons that TOM1953 said. Rubber does have air (beware of peroxide cures) trapped and air can get trapped inside your mould with the rubber. The smaller things are, the easy it is to not get air trapped, or to have all the air from inside the rubber released.
Asankaglove last doubt about the crosslinking and elongation is in fact not true. Elongation is not affected, for the stated principle of thickness, because of higher degree of crosslinking, but because there is more amount of material to distribute the aplied force. Of course that elongation is affected by crosslinking density! But not in terms of the same rubber but with different thicknesses.
Example: pick up an iron rod of the hardest you can find, with 1mm diameter, and say 1500mm length. Try to bend it by arm strenght only. You probably can. Now, for the same length and material, choose one with 100mm. Can you bend it (arm strenght again)? If you can, then superman is probably real. Chances are you can't.
Now, if the material is the same, the crystalline structure the same, with only changes in diameter, how can you bend one and not the other? Anwser: amount of material and not type of material through were your applied force is distributed. But you are right that elongation is affected by thickness. Not right as to the why.
Hope it helped!