Two possibilities occur to me, but I don't think much of either one, actually.

The first would be uneven moisture. I'm familiar with grape tomatoes splitting after a significant rainfall; I lose 10 - 25% or so. Perhaps you're not watering them enough. How dry is the soil a few inches down? Perhaps a really good soaking is necessary occasionally to keep enough moisture in the soil in your climate.

Theoretically, if you pick them after they've begun to turn yellow and let them ripen inside, there shouldn't be any splitting, as no additional moisture will get to the fruit. (Disclaimer: some small varieties are infamous for splitting when they're washed.) As long as they've already begun to turn color, being ripened off the plant should not affect the taste (there are other reasons why the horrible supermarket tomatoes taste like plastic).

==

Second possibility is that the tomatoes are actually ripe, and are splitting when they get beyond ripe.

Perhaps "ripe" is actually orange-red, and you're assuming ripe is a redder shade? It could even be -- you've got only one plant -- that your tomato was mis-labelled, and is actually a gold- or orange-fruited variety. The way to test this would be to pick some that are different shades of orange (lighter than the ones that split) and see how they taste. Also, take one that has split, trim it, and try the undamaged part: ripe, under-ripe, or over-ripe?

If the tomatoes that are splitting taste ripe or over-ripe, your tomato is not what you thought it was.

If the tomatoes that are splitting don't taste ripe, it's either a moisture problem or something I don't know about.

==

Additional information you could give us:

How are the tomatoes splitting? If it's standard radial splitting at the stem, the kind you see on many beefsteak varieties -- well, small fruit shouldn't split that way, and I wouldn't have a clue. If they're splitting irregularly, generally from the stem towards the blossom end, sometimes spiraling a bit -- that's the typical splitting caused by a sudden excess of moisture.

If you can't give us a photo, could you perhaps give us a link to a tomato picture which is the color at which your grapes split? These vendors, for instance, have plenty of pictures:
http://www.tomatogrowers.com/orange.htm -- this is the Orange page
http://mariseeds.com/seedcatalog/gallery.html -- all colors

What was the source of the plant? Did it have any other name besides "Grape"?

Is it growing in the ground or in a container? If a container, what is the size and the color (dark colors absorb more heat, and the plant can be unhappy, though I don't know if splitting would be a likely result)?

Are you using plenty of mulch around the plant? That cuts down on evaporation, and keeps the soil cooler too.

Are you growing any other tomato plants besides this grape? If so, what are they? Do any of them have problems with splitting?

Too much water. Tomatoes always split when overwatered.

I have a cherry tomato growing out the bottom of a 5 gallon hanging container. I only water it every other day. It does great on that watering schedule. So just cut back a bit. Try watering just once per day, and cut back until the leaves wilt slightly.

Sunny -
Thoroughly soak the area, mulch it heavily and then water enough to keep moisture constant under the mulch. Get and use a moisture meter.

I'm growing in hotter temps than you are (105+ for the past couple of weeks), and have not seen splitting on grape-size tomatoes.

The tiny ones, however, are shriveling into tom-aisins!

Yardenman - What works back east and what works in SoCal and the other arid parts of the country are entirely different. It's almost impossible to overwater.