Material to change its color by temperature condition meets over time There are thermochromic materials which their color changes reversible by temperature. Simply they show their current temperature by color. We can say their color is a function of current temperature:
$$c_t = f(temp_t)$$
Where $c_t$ is color in time $t$ and $temp_t$ is temperature in time $t$

I wonder if there is similar material which their color changes irreversible by history of temperature they meet over time. In comparison to thermochromic material, we can say their color changes by integrations of temperature:
$$c_t = f(\int_0^t temp_x dx)$$
Where $c_t$ is color in time $t$ and $temp_t$ is temperature in time $t$.

This seems to be more general and simpler than thermochromic materials as they are like regular chemical reactions such as oxidation, ... happens every day. 

Are there any industry suited solutions out there based on this behaviour which shown in the second equation?
Compounds that irreversibly record the thermal history of the surface they are attached do do exist and others are being developed.

One class, used in harsh industrial environments is described on the Sensor Coating Systems site.

There are chemical systems used to detect high temperatures in refrigerated food, but most don't seem to be single chemicals. There are certainly many devices in development, but how many involve single chemicals is unclear.