I live in Redwood City, Bay Area California (From wiki -Redwood City, along with most of the Bay Area, enjoys a mild Mediterranean climate, with warm, dry summers and cool, relatively wet winters.) and we have a morning glory in a potted container. We have had it from last summer. This summer it has developed brown spots on its leaves and some of the leaves are also yellowing. The brown spots are on both yellow and green leaves. I did some research but does not look like it is a insect problem. We have been watering it consistently allowing the soil to dry out.
Morning glories love sun and even high temperatures shouldn't cause the damage shown.
Have you added new soil or compost to the planter? Any slow release fertilizers? How regularly have you watered the soil in the planter? Not on a strict schedule but to make certain it's neither too dry nor not too wet.
Start by thoroughly cleaning up your planter, removing all dying and dead parts. By leaving it as you have, you're allowing any viral or fungal disease to continue spreading. If it's cultural mismanagement instead, you're still inviting fungal diseases to develop in the dead plant matter, especially if it gets wet.
Dig out the worse affected plant to check on the roots (or tuber depending on what species morning glory you have). Look for signs of root rot or black canker, if it's a tuber. If you find it present and if the soil hasn't been soggy through the winer and spring, it may be a fungal disease that attacks the roots. I've never found morning glories to be troubled by pests or diseases but I've only grown them as annuals.
If you haven't amended the soil since last year, do so and add a slow release fertilizer mixed in. Since I can only see part of the planter, I don't know its size. Using the same soil mix year after year without replenishing the soil is always bad practice. Since I can't examine your plants or soil mix, I can't say for certainty what the cause is. But I'm leaning toward depleted soil and/or inconsistent watering.