Eggplant, once established can withstand a greater drought than most vegetables. They are heat loving plants, but even heat loving plants require some water.
You will notice blossom drop and diminished yields when the daytime temperatures remain above 95 degrees for any extended period of time. The fruit may also develop sun-scald if exposed to excessive sunlight.
When night time temperatures are below 50 F they will not set fruit, however, they will set fruit in temperatures over 100 F. [with diminished yields] so long as they have moisture and nutrients.
Adequate moisture for eggplant is substantially lower than it is for other vegetables, in fact it should not be over watered as it is susceptible to root rot. Under normal conditions water sufficiently to moisten the soil to a depth of about 6 inches.
The critical period for moisture would be during fruit set and fruit development - excessive water at this time can hinder fruit development.
6 inches of Water ideal, but NOT 100% essential - they can do just fine with slightly less water under drought conditions.
Drought Tolerant Varieties
Some varieties of eggplants fare better under drought conditions than others. Some have genetic traits acquired via breeding or natural selection that make them more successful in dry hot conditions.
Ping Tung Eggplant is a drought resistant heirloom from Taiwan. The Fruits are long and purplish, about 2 inches in diameter and up to a foot and half long. Like many of the longer eggplant varieties it does well in dry hot conditions.
Listada de Gandia eggplant produces a 5 inch long Violet fruit with irregular white streaks, and grows very well under drought conditions. A similar drought tolerant variety is Rosa Bianca, a Sicilian Heirloom that reaches 4 to 6 inches in diameter and 5 to 7 inches in length.
Round Mauve - This is an early variety that will yield before the full force of summer heat bears down and is drought tolerant.
Diamond Eggplants are a Russian variety that are sometimes suggested as a drought resistant or tolerant variety. While it is true they are drought tolerant - they are not very heat tolerant and produce best in cooler ares and even higher elevations.
Florida High Bush, Pandora Striped Rose and Round Mauve are a few other eggplant cultivars that are drought tolerant.
Tips and Techniques
In addition to selecting the best varieties, suitable to your unique predicament, there are a number of fairly easy tactics to help eggplant prosper in droughts.
1. Consider drip irrigation - Drip irrigation uses use 30 - 50% less water than standard conventional watering methods, and uses it more effectively. See - Drip Irrigation for drought Gardening
2. Prepare your beds by adding large amounts of organic material and compost several inches deep. Work it into the soil well, it will help supply much needed nutrients to the plants as well as enhance the moisture retention quality of the soil
3. Add another 3 to 4 inches of organic mulch around the base of plants to help with moisture retention.
4. It is believed - although not proven beyond the shadow of a doubt that adding aspirin to the water when planting will enhance their ability to cope with drought as well as disease resistance. "The basic premise is that salicylic acid (the original active ingredient in aspirin) triggers plants to activate their version of an immune system, leading to an improved ability to fight off disease and fend off stresses like drought or cold."  .
5. Dusting the roots with Mychorizae is another well advised option when planting. Mychorizae are a naturally occurring beneficial fungi that attach themselves to plant roots creating a symbiotic relationship.
Mychorizae penetrate the root systems of most plants in nature. They are helpful in providing improved uptake of water and nutrients from the soil. They help protect the roots from harmful pathogens and disease and help to reduce the effects of mineral Build Up on plants.
6. Silicon is proven to enhance drought and heat tolerance in plants. No - I am not talking about putting silicone caulk on your plants ! Silicon is a very abundant natural element that is present in soil in fact it comprises nearly 1/4 of the Earths Crust. It strengthens the cellular walls, improving the plants tissue strength, overall health, vitality and productivity and is advantageous to drought tolerance in plants. It can be applied as foliar spray or as a soil addendum. See - Silicon for Drought Gardening
Note 1. Aspirin water for plant disease? Penn Live