It really depends if you wanted to depend on some specific 8051 variant vendor. It is best to use vendor provided registers, but if you changed vendor your code will brake, or even worse, it will misbehave.
If you had external RAM in your system (and it was not battery powered), than you could write a sequence of bytes (like 0xAA, 0x55...) somewhere in the reserved part of the memory, and check if it was still there after start up. If not, you have had a cold start. Of course, you should modify assembler start up code to make sure it does not initialize this part of memory (or it would be zero at each start), and you should instruct your linker to exclude this memory from linkage so that it does not get used by anything else.
Finally, include conditional compilation in your code so that if you had some 8051 variant with special registers, it would be used, if not, try the plan B.
I have done that with few bytes of internal 8051 memory /all my external RAM was battery powered/ and then I have learned than not every 8051 variant has had consistent policy at start up - some have all their internal memory initialized, some have initialized only SFR and some other specific areas leaving me few bytes to play with the procedure described.