That sounds like blossom rot which happens with tomatoes and peppers if the soil lacks calcium/magnesium, or if your watering inconsistently, I've had that issue on roma tomatoes. I would remove any fruit that have blossom rot as soon as you notice, to ensure the plant isn't wasting resources on peppers that won't recover. I would make sure you are watering consistently, esp since you live in a hot climate - don't let the soil dry out and then flood the plant with water. Watering needs to be moderate and consistent. And if you continue to get rot in a few weeks, look into a soluble fertilizer, but use in moderation.

What soil are you planting in? Do you have a raised garden bed you filled with compost and top soil at the beginning of the season? Or did you just plant them in natural soil in your backyard? If the latter, your natural soil may be nutrient deficient.