The e8500 is clocked higher and will generally run a bit hotter. 10c isn't that big of a difference and sounds about right. Tdp is just an estimated amount of heat in watts that cooling needs to meet or be capable of effectively removing in order to keep the cpu within operating temps. Just like the i5 4690k and i7 4790k are both 88w tdp cpu's yet due to hyper threading and increased clock speeds the i7 will generally run hotter.
If your cooling was just enough for the e5400 you may need a slightly better cooler for the e8500. 70c+ seems to be getting pretty warm, generally I'd prefer to be around 60-65c under full load on a core 2 duo.
Which cooler are you using? Is the cpu overclocked? What are the ambient room temps and is there decent case airflow? (ie, are there cluttered cables, dust and only 1 case fan or at least 1 intake and 1 exhaust with cables out of the way and thoroughly dusted out). All those things will impact cpu temps but doesn't mean it's fine for temps to keep running higher and higher.
As an example, say using the stock cooler with good airflow, ambient room temps around 22c you get 60c under load. 60c is a decent max load temp. Just because the stock cooler works well in that environment doesn't mean it's fine no matter what.
By contrast, say you're using the same stock cooler but the case airflow isn't ideal, there are cables in the way, dusty case and cpu cooler, ambient room temps are 30c and you're getting 80c under load. If dusting and cable management doesn't improve temps then you'd want to consider a better cooler to offset a poorly cooled case or higher ambient temps to get the cpu back closer to 60-65c. The environment it's running in and cooling conditions may create a need for better cooling solutions, as well as running a faster/hotter cpu compared to the older one.
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