Not really a problem. Almost all of my C172 hours were hot and high in South Africa where the density altitude can top 10000 feet on a summer afternoon. The farm strips are longer than those in the UK but in general the 172 behaves as per the POH provided you lean it properly before takeoff and pay attention to weight and balance.
Yes, you need more runway than at sea level but the information in the POH is accurate, you just have to pay attention to the limits.
The 172 was designed to carry 2 adults and 2 children, if you want more load and better short field performance rather use a 182.