Temperature is a smaller factor in reflux than acidity or fat content, but it does play a role, and the honest answer is that it depends on the person and the specific food.
Where cold can help
Cold water or a cold, non-acidic smoothie can feel soothing to an irritated throat after a reflux episode, similar to how a cold drink feels comforting with a sore throat from any cause. Very cold foods eaten slowly may also naturally slow down how quickly you're eating, which is itself a reflux-friendly habit.
Where cold can work against you
Very cold food or drinks consumed quickly, especially in large volumes, can cause the stomach to contract more forcefully, which in some people brings on symptoms rather than easing them. Ice cream is a common example where the cold temperature doesn't offset the high fat content — it's still likely to trigger symptoms despite feeling soothing in the moment.
A reasonable approach
If a cold drink after a spicy or acidic meal seems to help you specifically, there's little downside to continuing that habit. But it's not a reliable fix on its own, and it doesn't offset the impact of a genuinely triggering meal — think of it as a minor comfort measure, not a treatment.
