My understanding of the situation is that drinking a cold drink will have no beneficial effects on cooling you down. Though it may cause your blood vessels to tighten and thus restrict blood flow which then ultimately makes your body retain heat.
Looking at the science behind it, I found a study conducted A. R. Bain, N. C. Lesperance, and O. Jay which was published by the Acta Physiologica in 2012. The study came to the following conclusion:
**"Under conditions permitting full sweat evaporation, body heat storage is lower with warm water ingestion, kely because of disproportionate modulations in sweat output arising from warm‐sensitive thermosensors in the esophagus/stomach. Local temperature changes of the rectum following fluid ingestion exacerbate the previously identified error of thermometric heat storage estimations."
Another study came to a different but ultimately comparable result.
Although both groups significantly increased their core temperature (p<0.001) over the course of the exercise session and presented a significant decline in hydration status (p<0.001), participants in the COLD water trial had a significantly (p=0.024) smaller rise in core temperature (0.83°) over the duration of the trial in comparison to RT (1.13°). The participants in the COLD water trial were able to delay their increase in core body temperature for at least 30 minutes, whereas participants in the RT trial increased body temperature from baseline after 15 minutes. There was no significant difference between the COLD or the RT trials in broad jump and TTE performance tests. Bench press showed a small, albeit significant (p=0.046), decrease in performance when drinking COLD.
There are also articles out there which state the polar opposite. Though I am not so sure about the science behind them. What makes me sway in the other direction is that all cultures which are based in warm/hot climates eventually started drinking warm drinks and still do so.