I’m not fully sure of the context of your question. If I assume by the word ‘exist’ that you mean that is a ‘concrete noun’ (as opposed to an abstract noun); that it has a tangible reality beyond our collective imaginations.
Temperature and Time are both abstract nouns. So they only ‘exist’ in our collective minds.
Temperature is the calibration framework based on a standard set of units, by which we measure heat. Heat is real, temperature is abstract. Temperature is the dimension of heat (a dimension being an abstract calibration framework).
So no, temperature does not exist (beyond mans collective mind), it is heat that is real, that exists.
The word Time has two core meanings, two different contexts.
In one context Time too is, like Temperature, a dimension, i.e. an abstract calibration framework. Time is the dimension of change. Time is how we calibrate and index change. We say, for instance, motion (i.e. changing position ) is calibrated as, say, miles PER HOUR. The per hour bit is the calibration of the change. Time (in this context) is the dimension of change. Time is abstract, change is real.
So in this context Time, like temperature, only exists in our collective minds.
Time has another seperate (though similar) meaning. It is also a collective noun. It is a mass noun. Mass nouns are non-specific sets of something. For example ‘Traffic’ is a mass noun. Traffic is a non-specific set of vehicles. We give Traffic physical attributes…we say it stops, it starts, it flows, for example. But in reality its the underlying vehicles that move. Fundamentally Traffic doesn’t exist, its an abstract collective noun. Only individual vehicles exist.
Simarly, Time is a mass noun. It is a non-specific set of change-events. The only evidence of ‘Time passing’ is change/change-events. Time is to Change what Traffic is to vehicles. We give Time physical attributes…it starts, it stops, it flows. But fundamentally Time doesn’t exist, its an abstract collective noun. Only individual change-events exist.
But try telling a taxi driver late for the airport that Time and Traffic don’t exist..
What is Time? 